Properties

Degree 32
Conductor $ 2^{16} \cdot 3^{16} \cdot 5^{32} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + 4·5-s + 2·9-s + 2·11-s + 20·13-s + 16-s − 30·17-s + 8·20-s − 10·23-s + 20·25-s − 10·29-s − 18·31-s + 4·36-s + 20·37-s + 22·41-s + 4·44-s + 8·45-s − 50·47-s + 30·49-s + 40·52-s + 30·53-s + 8·55-s + 20·59-s + 12·61-s + 80·65-s − 50·67-s − 60·68-s + ⋯
L(s)  = 1  + 4-s + 1.78·5-s + 2/3·9-s + 0.603·11-s + 5.54·13-s + 1/4·16-s − 7.27·17-s + 1.78·20-s − 2.08·23-s + 4·25-s − 1.85·29-s − 3.23·31-s + 2/3·36-s + 3.28·37-s + 3.43·41-s + 0.603·44-s + 1.19·45-s − 7.29·47-s + 30/7·49-s + 5.54·52-s + 4.12·53-s + 1.07·55-s + 2.60·59-s + 1.53·61-s + 9.92·65-s − 6.10·67-s − 7.27·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{16} \cdot 3^{16} \cdot 5^{32}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(32,\ 2^{16} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )$
$L(1)$  $\approx$  $4.48459$
$L(\frac12)$  $\approx$  $4.48459$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
3 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
5 \( 1 - 4 T - 4 T^{2} + 6 T^{3} + 161 T^{4} - 52 p T^{5} - 116 p T^{6} - 104 p T^{7} + 1441 p T^{8} - 104 p^{2} T^{9} - 116 p^{3} T^{10} - 52 p^{4} T^{11} + 161 p^{4} T^{12} + 6 p^{5} T^{13} - 4 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
good7 \( 1 - 30 T^{2} + 527 T^{4} - 6940 T^{6} + 74703 T^{8} - 716280 T^{10} + 6275029 T^{12} - 50516750 T^{14} + 372187880 T^{16} - 50516750 p^{2} T^{18} + 6275029 p^{4} T^{20} - 716280 p^{6} T^{22} + 74703 p^{8} T^{24} - 6940 p^{10} T^{26} + 527 p^{12} T^{28} - 30 p^{14} T^{30} + p^{16} T^{32} \)
11 \( 1 - 2 T + p T^{2} - 50 T^{3} + 270 T^{4} + 764 T^{5} - 168 T^{6} + 4404 T^{7} - 16480 T^{8} + 222470 T^{9} + 41985 p T^{10} - 66610 T^{11} + 2053685 T^{12} + 10233320 T^{13} + 85541270 T^{14} + 166971000 T^{15} - 141281840 T^{16} + 166971000 p T^{17} + 85541270 p^{2} T^{18} + 10233320 p^{3} T^{19} + 2053685 p^{4} T^{20} - 66610 p^{5} T^{21} + 41985 p^{7} T^{22} + 222470 p^{7} T^{23} - 16480 p^{8} T^{24} + 4404 p^{9} T^{25} - 168 p^{10} T^{26} + 764 p^{11} T^{27} + 270 p^{12} T^{28} - 50 p^{13} T^{29} + p^{15} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \)
13 \( 1 - 20 T + 240 T^{2} - 2100 T^{3} + 14722 T^{4} - 86440 T^{5} + 437000 T^{6} - 1922660 T^{7} + 7352283 T^{8} - 23888120 T^{9} + 61481440 T^{10} - 91205000 T^{11} - 205909876 T^{12} + 2578228580 T^{13} - 14615222400 T^{14} + 65003714260 T^{15} - 249510986035 T^{16} + 65003714260 p T^{17} - 14615222400 p^{2} T^{18} + 2578228580 p^{3} T^{19} - 205909876 p^{4} T^{20} - 91205000 p^{5} T^{21} + 61481440 p^{6} T^{22} - 23888120 p^{7} T^{23} + 7352283 p^{8} T^{24} - 1922660 p^{9} T^{25} + 437000 p^{10} T^{26} - 86440 p^{11} T^{27} + 14722 p^{12} T^{28} - 2100 p^{13} T^{29} + 240 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} \)
17 \( 1 + 30 T + 500 T^{2} + 5990 T^{3} + 56782 T^{4} + 447460 T^{5} + 177380 p T^{6} + 17669550 T^{7} + 90820363 T^{8} + 409992380 T^{9} + 1611336300 T^{10} + 5357847740 T^{11} + 13820507184 T^{12} + 18014105810 T^{13} - 71676889400 T^{14} - 711604275870 T^{15} - 3544790874535 T^{16} - 711604275870 p T^{17} - 71676889400 p^{2} T^{18} + 18014105810 p^{3} T^{19} + 13820507184 p^{4} T^{20} + 5357847740 p^{5} T^{21} + 1611336300 p^{6} T^{22} + 409992380 p^{7} T^{23} + 90820363 p^{8} T^{24} + 17669550 p^{9} T^{25} + 177380 p^{11} T^{26} + 447460 p^{11} T^{27} + 56782 p^{12} T^{28} + 5990 p^{13} T^{29} + 500 p^{14} T^{30} + 30 p^{15} T^{31} + p^{16} T^{32} \)
19 \( 1 + 24 T^{2} - 30 T^{3} + 330 T^{4} + 740 T^{5} + 7630 T^{6} + 990 p T^{7} + 160255 T^{8} - 244020 T^{9} + 7105152 T^{10} - 5384250 T^{11} + 141174008 T^{12} - 174364190 T^{13} + 1779959840 T^{14} + 4165821270 T^{15} + 38170716365 T^{16} + 4165821270 p T^{17} + 1779959840 p^{2} T^{18} - 174364190 p^{3} T^{19} + 141174008 p^{4} T^{20} - 5384250 p^{5} T^{21} + 7105152 p^{6} T^{22} - 244020 p^{7} T^{23} + 160255 p^{8} T^{24} + 990 p^{10} T^{25} + 7630 p^{10} T^{26} + 740 p^{11} T^{27} + 330 p^{12} T^{28} - 30 p^{13} T^{29} + 24 p^{14} T^{30} + p^{16} T^{32} \)
23 \( 1 + 10 T + 30 T^{2} - 250 T^{3} - 2718 T^{4} - 8680 T^{5} + 13790 T^{6} + 227450 T^{7} + 942143 T^{8} + 3042820 T^{9} + 4039430 T^{10} - 83513580 T^{11} - 909320196 T^{12} - 3918445730 T^{13} - 578679600 T^{14} + 92103435490 T^{15} + 633292773765 T^{16} + 92103435490 p T^{17} - 578679600 p^{2} T^{18} - 3918445730 p^{3} T^{19} - 909320196 p^{4} T^{20} - 83513580 p^{5} T^{21} + 4039430 p^{6} T^{22} + 3042820 p^{7} T^{23} + 942143 p^{8} T^{24} + 227450 p^{9} T^{25} + 13790 p^{10} T^{26} - 8680 p^{11} T^{27} - 2718 p^{12} T^{28} - 250 p^{13} T^{29} + 30 p^{14} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \)
29 \( 1 + 10 T - 36 T^{2} - 270 T^{3} + 3910 T^{4} + 4460 T^{5} - 185440 T^{6} + 333090 T^{7} + 4522435 T^{8} - 38473780 T^{9} - 69131068 T^{10} + 1395128220 T^{11} - 3409890032 T^{12} - 36344819210 T^{13} + 245632726680 T^{14} + 543098432830 T^{15} - 7660398480495 T^{16} + 543098432830 p T^{17} + 245632726680 p^{2} T^{18} - 36344819210 p^{3} T^{19} - 3409890032 p^{4} T^{20} + 1395128220 p^{5} T^{21} - 69131068 p^{6} T^{22} - 38473780 p^{7} T^{23} + 4522435 p^{8} T^{24} + 333090 p^{9} T^{25} - 185440 p^{10} T^{26} + 4460 p^{11} T^{27} + 3910 p^{12} T^{28} - 270 p^{13} T^{29} - 36 p^{14} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \)
31 \( 1 + 18 T + 121 T^{2} + 610 T^{3} + 6060 T^{4} + 54724 T^{5} + 309962 T^{6} + 1456784 T^{7} + 9332270 T^{8} + 64865970 T^{9} + 327433735 T^{10} + 1367357590 T^{11} + 7702756335 T^{12} + 47160385620 T^{13} + 227624335970 T^{14} + 1111476930100 T^{15} + 6249005152660 T^{16} + 1111476930100 p T^{17} + 227624335970 p^{2} T^{18} + 47160385620 p^{3} T^{19} + 7702756335 p^{4} T^{20} + 1367357590 p^{5} T^{21} + 327433735 p^{6} T^{22} + 64865970 p^{7} T^{23} + 9332270 p^{8} T^{24} + 1456784 p^{9} T^{25} + 309962 p^{10} T^{26} + 54724 p^{11} T^{27} + 6060 p^{12} T^{28} + 610 p^{13} T^{29} + 121 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \)
37 \( 1 - 20 T + 265 T^{2} - 2300 T^{3} + 17532 T^{4} - 115640 T^{5} + 825500 T^{6} - 5506160 T^{7} + 38914148 T^{8} - 253492260 T^{9} + 1794329215 T^{10} - 11470149820 T^{11} + 74232183599 T^{12} - 424648096560 T^{13} + 2693507120800 T^{14} - 16225080394680 T^{15} + 105486161388440 T^{16} - 16225080394680 p T^{17} + 2693507120800 p^{2} T^{18} - 424648096560 p^{3} T^{19} + 74232183599 p^{4} T^{20} - 11470149820 p^{5} T^{21} + 1794329215 p^{6} T^{22} - 253492260 p^{7} T^{23} + 38914148 p^{8} T^{24} - 5506160 p^{9} T^{25} + 825500 p^{10} T^{26} - 115640 p^{11} T^{27} + 17532 p^{12} T^{28} - 2300 p^{13} T^{29} + 265 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} \)
41 \( 1 - 22 T + 316 T^{2} - 3710 T^{3} + 39530 T^{4} - 407436 T^{5} + 3918652 T^{6} - 35382126 T^{7} + 301988495 T^{8} - 2475859380 T^{9} + 19707200260 T^{10} - 150750040860 T^{11} + 1111070917960 T^{12} - 7891095298530 T^{13} + 54497096107120 T^{14} - 366799083339050 T^{15} + 2387443151993985 T^{16} - 366799083339050 p T^{17} + 54497096107120 p^{2} T^{18} - 7891095298530 p^{3} T^{19} + 1111070917960 p^{4} T^{20} - 150750040860 p^{5} T^{21} + 19707200260 p^{6} T^{22} - 2475859380 p^{7} T^{23} + 301988495 p^{8} T^{24} - 35382126 p^{9} T^{25} + 3918652 p^{10} T^{26} - 407436 p^{11} T^{27} + 39530 p^{12} T^{28} - 3710 p^{13} T^{29} + 316 p^{14} T^{30} - 22 p^{15} T^{31} + p^{16} T^{32} \)
43 \( 1 - 360 T^{2} + 63972 T^{4} - 7536360 T^{6} + 666842308 T^{8} - 47508203160 T^{10} + 2846509586204 T^{12} - 147613243096600 T^{14} + 6742598283189430 T^{16} - 147613243096600 p^{2} T^{18} + 2846509586204 p^{4} T^{20} - 47508203160 p^{6} T^{22} + 666842308 p^{8} T^{24} - 7536360 p^{10} T^{26} + 63972 p^{12} T^{28} - 360 p^{14} T^{30} + p^{16} T^{32} \)
47 \( 1 + 50 T + 1290 T^{2} + 23780 T^{3} + 355482 T^{4} + 4545300 T^{5} + 51239280 T^{6} + 520209820 T^{7} + 4818856003 T^{8} + 41087515780 T^{9} + 324882626890 T^{10} + 2399419588970 T^{11} + 16710816340744 T^{12} + 111327617454640 T^{13} + 724257830956800 T^{14} + 4728319081214120 T^{15} + 31780196573425365 T^{16} + 4728319081214120 p T^{17} + 724257830956800 p^{2} T^{18} + 111327617454640 p^{3} T^{19} + 16710816340744 p^{4} T^{20} + 2399419588970 p^{5} T^{21} + 324882626890 p^{6} T^{22} + 41087515780 p^{7} T^{23} + 4818856003 p^{8} T^{24} + 520209820 p^{9} T^{25} + 51239280 p^{10} T^{26} + 4545300 p^{11} T^{27} + 355482 p^{12} T^{28} + 23780 p^{13} T^{29} + 1290 p^{14} T^{30} + 50 p^{15} T^{31} + p^{16} T^{32} \)
53 \( 1 - 30 T + 655 T^{2} - 10630 T^{3} + 152332 T^{4} - 1919860 T^{5} + 22723040 T^{6} - 248684220 T^{7} + 2591225028 T^{8} - 25379094630 T^{9} + 238630725505 T^{10} - 2134501287330 T^{11} + 18442776242119 T^{12} - 152410720886420 T^{13} + 1218238235823200 T^{14} - 9336287074370240 T^{15} + 69349166231444240 T^{16} - 9336287074370240 p T^{17} + 1218238235823200 p^{2} T^{18} - 152410720886420 p^{3} T^{19} + 18442776242119 p^{4} T^{20} - 2134501287330 p^{5} T^{21} + 238630725505 p^{6} T^{22} - 25379094630 p^{7} T^{23} + 2591225028 p^{8} T^{24} - 248684220 p^{9} T^{25} + 22723040 p^{10} T^{26} - 1919860 p^{11} T^{27} + 152332 p^{12} T^{28} - 10630 p^{13} T^{29} + 655 p^{14} T^{30} - 30 p^{15} T^{31} + p^{16} T^{32} \)
59 \( 1 - 20 T - 111 T^{2} + 3640 T^{3} + 11660 T^{4} - 317820 T^{5} - 1817980 T^{6} + 11679820 T^{7} + 217773860 T^{8} + 627241760 T^{9} - 13630850673 T^{10} - 133224376740 T^{11} + 186113511463 T^{12} + 9371240514920 T^{13} + 48131664631280 T^{14} - 252534601553160 T^{15} - 4526062111550040 T^{16} - 252534601553160 p T^{17} + 48131664631280 p^{2} T^{18} + 9371240514920 p^{3} T^{19} + 186113511463 p^{4} T^{20} - 133224376740 p^{5} T^{21} - 13630850673 p^{6} T^{22} + 627241760 p^{7} T^{23} + 217773860 p^{8} T^{24} + 11679820 p^{9} T^{25} - 1817980 p^{10} T^{26} - 317820 p^{11} T^{27} + 11660 p^{12} T^{28} + 3640 p^{13} T^{29} - 111 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} \)
61 \( 1 - 12 T - 104 T^{2} + 1460 T^{3} + 9310 T^{4} - 110696 T^{5} - 1128308 T^{6} + 9605804 T^{7} + 95707895 T^{8} - 545482080 T^{9} - 8568890440 T^{10} + 31012865040 T^{11} + 647686771260 T^{12} - 1172976299580 T^{13} - 45238788397880 T^{14} + 2951178504100 T^{15} + 3155660575885285 T^{16} + 2951178504100 p T^{17} - 45238788397880 p^{2} T^{18} - 1172976299580 p^{3} T^{19} + 647686771260 p^{4} T^{20} + 31012865040 p^{5} T^{21} - 8568890440 p^{6} T^{22} - 545482080 p^{7} T^{23} + 95707895 p^{8} T^{24} + 9605804 p^{9} T^{25} - 1128308 p^{10} T^{26} - 110696 p^{11} T^{27} + 9310 p^{12} T^{28} + 1460 p^{13} T^{29} - 104 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \)
67 \( 1 + 50 T + 1390 T^{2} + 30100 T^{3} + 553682 T^{4} + 8918100 T^{5} + 129754300 T^{6} + 1739003620 T^{7} + 21685053163 T^{8} + 253853201940 T^{9} + 2808527300190 T^{10} + 440074822710 p T^{11} + 294663718701784 T^{12} + 2810511152550240 T^{13} + 25630455288876000 T^{14} + 223725991474439120 T^{15} + 1870714073010261365 T^{16} + 223725991474439120 p T^{17} + 25630455288876000 p^{2} T^{18} + 2810511152550240 p^{3} T^{19} + 294663718701784 p^{4} T^{20} + 440074822710 p^{6} T^{21} + 2808527300190 p^{6} T^{22} + 253853201940 p^{7} T^{23} + 21685053163 p^{8} T^{24} + 1739003620 p^{9} T^{25} + 129754300 p^{10} T^{26} + 8918100 p^{11} T^{27} + 553682 p^{12} T^{28} + 30100 p^{13} T^{29} + 1390 p^{14} T^{30} + 50 p^{15} T^{31} + p^{16} T^{32} \)
71 \( 1 + 28 T + 256 T^{2} + 1150 T^{3} + 20770 T^{4} + 328604 T^{5} + 2351902 T^{6} + 21115094 T^{7} + 321977895 T^{8} + 2983992020 T^{9} + 18685321160 T^{10} + 180970630790 T^{11} + 2127756781760 T^{12} + 17298444226470 T^{13} + 123520240637920 T^{14} + 1239239342309850 T^{15} + 12274938369025885 T^{16} + 1239239342309850 p T^{17} + 123520240637920 p^{2} T^{18} + 17298444226470 p^{3} T^{19} + 2127756781760 p^{4} T^{20} + 180970630790 p^{5} T^{21} + 18685321160 p^{6} T^{22} + 2983992020 p^{7} T^{23} + 321977895 p^{8} T^{24} + 21115094 p^{9} T^{25} + 2351902 p^{10} T^{26} + 328604 p^{11} T^{27} + 20770 p^{12} T^{28} + 1150 p^{13} T^{29} + 256 p^{14} T^{30} + 28 p^{15} T^{31} + p^{16} T^{32} \)
73 \( 1 - 20 T + 380 T^{2} - 7420 T^{3} + 107282 T^{4} - 1509240 T^{5} + 20338260 T^{6} - 240254580 T^{7} + 2794851843 T^{8} - 31155740720 T^{9} + 320911861980 T^{10} - 3274927660920 T^{11} + 32181004976604 T^{12} - 299417039187980 T^{13} + 2775262759264800 T^{14} - 24764197420905260 T^{15} + 211758644546106765 T^{16} - 24764197420905260 p T^{17} + 2775262759264800 p^{2} T^{18} - 299417039187980 p^{3} T^{19} + 32181004976604 p^{4} T^{20} - 3274927660920 p^{5} T^{21} + 320911861980 p^{6} T^{22} - 31155740720 p^{7} T^{23} + 2794851843 p^{8} T^{24} - 240254580 p^{9} T^{25} + 20338260 p^{10} T^{26} - 1509240 p^{11} T^{27} + 107282 p^{12} T^{28} - 7420 p^{13} T^{29} + 380 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} \)
79 \( 1 + 20 T + 224 T^{2} + 1780 T^{3} + 6070 T^{4} + 5760 T^{5} + 11080 p T^{6} + 20333540 T^{7} + 305719495 T^{8} + 3198483920 T^{9} + 22059896192 T^{10} + 116979470440 T^{11} + 8355284812 p T^{12} + 7893347694940 T^{13} + 139701887310360 T^{14} + 1828379420794980 T^{15} + 18018691485968485 T^{16} + 1828379420794980 p T^{17} + 139701887310360 p^{2} T^{18} + 7893347694940 p^{3} T^{19} + 8355284812 p^{5} T^{20} + 116979470440 p^{5} T^{21} + 22059896192 p^{6} T^{22} + 3198483920 p^{7} T^{23} + 305719495 p^{8} T^{24} + 20333540 p^{9} T^{25} + 11080 p^{11} T^{26} + 5760 p^{11} T^{27} + 6070 p^{12} T^{28} + 1780 p^{13} T^{29} + 224 p^{14} T^{30} + 20 p^{15} T^{31} + p^{16} T^{32} \)
83 \( 1 + 30 T + 535 T^{2} + 5780 T^{3} + 43882 T^{4} + 2470 p T^{5} + 590510 T^{6} + 2468290 T^{7} + 123659138 T^{8} + 2415976440 T^{9} + 27231081235 T^{10} + 158815212750 T^{11} + 405339323209 T^{12} + 3244201552980 T^{13} + 126271405899100 T^{14} + 2149314211865960 T^{15} + 23236430083058540 T^{16} + 2149314211865960 p T^{17} + 126271405899100 p^{2} T^{18} + 3244201552980 p^{3} T^{19} + 405339323209 p^{4} T^{20} + 158815212750 p^{5} T^{21} + 27231081235 p^{6} T^{22} + 2415976440 p^{7} T^{23} + 123659138 p^{8} T^{24} + 2468290 p^{9} T^{25} + 590510 p^{10} T^{26} + 2470 p^{12} T^{27} + 43882 p^{12} T^{28} + 5780 p^{13} T^{29} + 535 p^{14} T^{30} + 30 p^{15} T^{31} + p^{16} T^{32} \)
89 \( 1 - 70 T + 2119 T^{2} - 35330 T^{3} + 343460 T^{4} - 2032360 T^{5} + 10870620 T^{6} - 64684440 T^{7} - 584089940 T^{8} + 17002587830 T^{9} - 99745085123 T^{10} - 366225143690 T^{11} + 5693869232523 T^{12} - 24776539056740 T^{13} + 308063609313680 T^{14} - 3148141734399680 T^{15} + 20769024208268760 T^{16} - 3148141734399680 p T^{17} + 308063609313680 p^{2} T^{18} - 24776539056740 p^{3} T^{19} + 5693869232523 p^{4} T^{20} - 366225143690 p^{5} T^{21} - 99745085123 p^{6} T^{22} + 17002587830 p^{7} T^{23} - 584089940 p^{8} T^{24} - 64684440 p^{9} T^{25} + 10870620 p^{10} T^{26} - 2032360 p^{11} T^{27} + 343460 p^{12} T^{28} - 35330 p^{13} T^{29} + 2119 p^{14} T^{30} - 70 p^{15} T^{31} + p^{16} T^{32} \)
97 \( 1 + 10 T + 205 T^{2} + 4320 T^{3} + 44952 T^{4} + 667870 T^{5} + 8897460 T^{6} + 68756910 T^{7} + 834554408 T^{8} + 7477618800 T^{9} + 22708693755 T^{10} + 118605463990 T^{11} - 2491804851081 T^{12} - 84263627884560 T^{13} - 895266623177600 T^{14} - 11056342779615180 T^{15} - 138673183182585960 T^{16} - 11056342779615180 p T^{17} - 895266623177600 p^{2} T^{18} - 84263627884560 p^{3} T^{19} - 2491804851081 p^{4} T^{20} + 118605463990 p^{5} T^{21} + 22708693755 p^{6} T^{22} + 7477618800 p^{7} T^{23} + 834554408 p^{8} T^{24} + 68756910 p^{9} T^{25} + 8897460 p^{10} T^{26} + 667870 p^{11} T^{27} + 44952 p^{12} T^{28} + 4320 p^{13} T^{29} + 205 p^{14} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.97036263420644048569315890827, −3.96647364997502156248679473321, −3.85884944433142228619913926406, −3.69778008129017253310306431079, −3.57941304093049904094490642720, −3.33495566488987431538765481070, −3.22704791489889225024808475536, −3.15395405445264713373292220980, −3.15059353067334317615240501357, −3.02998996966798272484720143825, −3.00502029327306916201184719055, −2.56795626813200928523395999178, −2.42022892764829301230525190014, −2.41630060558005889507551185396, −2.32219329465399738568902724861, −2.28465551055071685817340172735, −2.08030982605615545982831490121, −2.00379005318336433334260222751, −1.87744328217536809060412156640, −1.82178062140016157885121275801, −1.65590536246921835616227913356, −1.43813782509858404915512192485, −1.23334448227241173793524948406, −1.00099931379038721721612524985, −0.72458716210564977758539970976, 0.72458716210564977758539970976, 1.00099931379038721721612524985, 1.23334448227241173793524948406, 1.43813782509858404915512192485, 1.65590536246921835616227913356, 1.82178062140016157885121275801, 1.87744328217536809060412156640, 2.00379005318336433334260222751, 2.08030982605615545982831490121, 2.28465551055071685817340172735, 2.32219329465399738568902724861, 2.41630060558005889507551185396, 2.42022892764829301230525190014, 2.56795626813200928523395999178, 3.00502029327306916201184719055, 3.02998996966798272484720143825, 3.15059353067334317615240501357, 3.15395405445264713373292220980, 3.22704791489889225024808475536, 3.33495566488987431538765481070, 3.57941304093049904094490642720, 3.69778008129017253310306431079, 3.85884944433142228619913926406, 3.96647364997502156248679473321, 3.97036263420644048569315890827

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.