Properties

Degree 16
Conductor $ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 32·4-s + 80·8-s − 8·11-s + 8·13-s + 120·16-s − 32·17-s − 64·22-s − 40·23-s − 24·25-s + 64·26-s + 144·31-s + 32·32-s − 256·34-s + 160·37-s − 320·41-s − 32·43-s − 256·44-s − 320·46-s − 144·47-s − 192·50-s + 256·52-s − 200·53-s + 288·61-s + 1.15e3·62-s − 384·64-s + 80·67-s + ⋯
L(s)  = 1  + 4·2-s + 8·4-s + 10·8-s − 0.727·11-s + 8/13·13-s + 15/2·16-s − 1.88·17-s − 2.90·22-s − 1.73·23-s − 0.959·25-s + 2.46·26-s + 4.64·31-s + 32-s − 7.52·34-s + 4.32·37-s − 7.80·41-s − 0.744·43-s − 5.81·44-s − 6.95·46-s − 3.06·47-s − 3.83·50-s + 4.92·52-s − 3.77·53-s + 4.72·61-s + 18.5·62-s − 6·64-s + 1.19·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{210} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.00229517\)
\(L(\frac12)\)  \(\approx\)  \(0.00229517\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( ( 1 - p T + p T^{2} )^{4} \)
3 \( ( 1 + p^{2} T^{4} )^{2} \)
5 \( 1 + 24 T^{2} + 2 p^{2} T^{4} + 24 p^{4} T^{6} + p^{8} T^{8} \)
7 \( ( 1 + p^{2} T^{4} )^{2} \)
good11 \( ( 1 + 4 T + 120 T^{2} - 964 T^{3} + 9758 T^{4} - 964 p^{2} T^{5} + 120 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
13 \( 1 - 8 T + 32 T^{2} - 1000 T^{3} + 57188 T^{4} - 379192 T^{5} + 10080 p^{2} T^{6} - 4420344 p T^{7} + 1905455878 T^{8} - 4420344 p^{3} T^{9} + 10080 p^{6} T^{10} - 379192 p^{6} T^{11} + 57188 p^{8} T^{12} - 1000 p^{10} T^{13} + 32 p^{12} T^{14} - 8 p^{14} T^{15} + p^{16} T^{16} \)
17 \( 1 + 32 T + 512 T^{2} + 13280 T^{3} + 494308 T^{4} + 9125408 T^{5} + 127106560 T^{6} + 2946229728 T^{7} + 67506528198 T^{8} + 2946229728 p^{2} T^{9} + 127106560 p^{4} T^{10} + 9125408 p^{6} T^{11} + 494308 p^{8} T^{12} + 13280 p^{10} T^{13} + 512 p^{12} T^{14} + 32 p^{14} T^{15} + p^{16} T^{16} \)
19 \( 1 - 1632 T^{2} + 1354756 T^{4} - 767600928 T^{6} + 320946269766 T^{8} - 767600928 p^{4} T^{10} + 1354756 p^{8} T^{12} - 1632 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 + 40 T + 800 T^{2} + 23320 T^{3} + 928156 T^{4} + 22803400 T^{5} + 441522400 T^{6} + 12864455160 T^{7} + 373563445830 T^{8} + 12864455160 p^{2} T^{9} + 441522400 p^{4} T^{10} + 22803400 p^{6} T^{11} + 928156 p^{8} T^{12} + 23320 p^{10} T^{13} + 800 p^{12} T^{14} + 40 p^{14} T^{15} + p^{16} T^{16} \)
29 \( 1 - 5816 T^{2} + 15494364 T^{4} - 24578727304 T^{6} + 25344087778310 T^{8} - 24578727304 p^{4} T^{10} + 15494364 p^{8} T^{12} - 5816 p^{12} T^{14} + p^{16} T^{16} \)
31 \( ( 1 - 72 T + 4816 T^{2} - 187848 T^{3} + 7076706 T^{4} - 187848 p^{2} T^{5} + 4816 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( 1 - 160 T + 12800 T^{2} - 692576 T^{3} + 28393916 T^{4} - 973559072 T^{5} + 32158084608 T^{6} - 1118111280864 T^{7} + 40611983255110 T^{8} - 1118111280864 p^{2} T^{9} + 32158084608 p^{4} T^{10} - 973559072 p^{6} T^{11} + 28393916 p^{8} T^{12} - 692576 p^{10} T^{13} + 12800 p^{12} T^{14} - 160 p^{14} T^{15} + p^{16} T^{16} \)
41 \( ( 1 + 160 T + 11944 T^{2} + 591520 T^{3} + 24800530 T^{4} + 591520 p^{2} T^{5} + 11944 p^{4} T^{6} + 160 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
43 \( 1 + 32 T + 512 T^{2} - 44896 T^{3} - 2147132 T^{4} + 116117984 T^{5} + 5822932480 T^{6} + 160684182624 T^{7} + 3183857489478 T^{8} + 160684182624 p^{2} T^{9} + 5822932480 p^{4} T^{10} + 116117984 p^{6} T^{11} - 2147132 p^{8} T^{12} - 44896 p^{10} T^{13} + 512 p^{12} T^{14} + 32 p^{14} T^{15} + p^{16} T^{16} \)
47 \( 1 + 144 T + 10368 T^{2} + 613776 T^{3} + 40380676 T^{4} + 2550145200 T^{5} + 136914549120 T^{6} + 6692132733360 T^{7} + 317421876290310 T^{8} + 6692132733360 p^{2} T^{9} + 136914549120 p^{4} T^{10} + 2550145200 p^{6} T^{11} + 40380676 p^{8} T^{12} + 613776 p^{10} T^{13} + 10368 p^{12} T^{14} + 144 p^{14} T^{15} + p^{16} T^{16} \)
53 \( 1 + 200 T + 20000 T^{2} + 1719464 T^{3} + 146284516 T^{4} + 10449435128 T^{5} + 642474929248 T^{6} + 39041749119576 T^{7} + 2230315686063750 T^{8} + 39041749119576 p^{2} T^{9} + 642474929248 p^{4} T^{10} + 10449435128 p^{6} T^{11} + 146284516 p^{8} T^{12} + 1719464 p^{10} T^{13} + 20000 p^{12} T^{14} + 200 p^{14} T^{15} + p^{16} T^{16} \)
59 \( 1 - 616 T^{2} + 1419676 T^{4} + 5207600552 T^{6} - 204890323221242 T^{8} + 5207600552 p^{4} T^{10} + 1419676 p^{8} T^{12} - 616 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 - 144 T + 14124 T^{2} - 1141872 T^{3} + 82373126 T^{4} - 1141872 p^{2} T^{5} + 14124 p^{4} T^{6} - 144 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( 1 - 80 T + 3200 T^{2} - 105296 T^{3} - 42100604 T^{4} + 2558251408 T^{5} - 64394556032 T^{6} - 2106545111664 T^{7} + 886628492165190 T^{8} - 2106545111664 p^{2} T^{9} - 64394556032 p^{4} T^{10} + 2558251408 p^{6} T^{11} - 42100604 p^{8} T^{12} - 105296 p^{10} T^{13} + 3200 p^{12} T^{14} - 80 p^{14} T^{15} + p^{16} T^{16} \)
71 \( ( 1 + 140 T + 26424 T^{2} + 2214100 T^{3} + 216062270 T^{4} + 2214100 p^{2} T^{5} + 26424 p^{4} T^{6} + 140 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
73 \( 1 - 312 T + 48672 T^{2} - 5493144 T^{3} + 450793636 T^{4} - 23206886280 T^{5} + 386836170336 T^{6} + 68692339480344 T^{7} - 8375186112407610 T^{8} + 68692339480344 p^{2} T^{9} + 386836170336 p^{4} T^{10} - 23206886280 p^{6} T^{11} + 450793636 p^{8} T^{12} - 5493144 p^{10} T^{13} + 48672 p^{12} T^{14} - 312 p^{14} T^{15} + p^{16} T^{16} \)
79 \( 1 - 2856 T^{2} - 59627684 T^{4} + 132589015272 T^{6} + 2412904555054278 T^{8} + 132589015272 p^{4} T^{10} - 59627684 p^{8} T^{12} - 2856 p^{12} T^{14} + p^{16} T^{16} \)
83 \( 1 + 320 T + 51200 T^{2} + 5915840 T^{3} + 553217092 T^{4} + 45528000320 T^{5} + 3742826444800 T^{6} + 317655699460800 T^{7} + 26711748349549254 T^{8} + 317655699460800 p^{2} T^{9} + 3742826444800 p^{4} T^{10} + 45528000320 p^{6} T^{11} + 553217092 p^{8} T^{12} + 5915840 p^{10} T^{13} + 51200 p^{12} T^{14} + 320 p^{14} T^{15} + p^{16} T^{16} \)
89 \( 1 - 26800 T^{2} + 323387236 T^{4} - 2700009379600 T^{6} + 20962417894146886 T^{8} - 2700009379600 p^{4} T^{10} + 323387236 p^{8} T^{12} - 26800 p^{12} T^{14} + p^{16} T^{16} \)
97 \( 1 + 24 T + 288 T^{2} + 197496 T^{3} - 223582556 T^{4} - 4223145048 T^{5} - 17461370016 T^{6} + 856985885448 T^{7} + 24034841148321606 T^{8} + 856985885448 p^{2} T^{9} - 17461370016 p^{4} T^{10} - 4223145048 p^{6} T^{11} - 223582556 p^{8} T^{12} + 197496 p^{10} T^{13} + 288 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.12015398653510732720568641796, −5.09677434211087661219120649232, −5.08969736985828003556101798591, −5.00673082250196025744122644956, −4.54856793771602274289158255176, −4.53720965940148724811634649469, −4.49094011885412783433562128268, −4.30641664752861511289395646882, −4.24678752068275589752388544941, −4.01753136566828914490718200427, −3.67322233830124865344256363194, −3.44562576555606804460360603355, −3.43449695687363813033309195095, −3.36065343991940842874867230240, −3.05700146024233257470936507950, −3.04183722672410897991439383008, −2.69715732058352496712588823737, −2.36386048854697758808198580105, −2.35593389668458868843520291710, −2.00239829246164356704608878141, −1.98634681396801574477360899676, −1.52241950878432744418559095692, −1.30810384807650183276600907175, −0.67161952533483485419618285373, −0.00275135664856288898731360347, 0.00275135664856288898731360347, 0.67161952533483485419618285373, 1.30810384807650183276600907175, 1.52241950878432744418559095692, 1.98634681396801574477360899676, 2.00239829246164356704608878141, 2.35593389668458868843520291710, 2.36386048854697758808198580105, 2.69715732058352496712588823737, 3.04183722672410897991439383008, 3.05700146024233257470936507950, 3.36065343991940842874867230240, 3.43449695687363813033309195095, 3.44562576555606804460360603355, 3.67322233830124865344256363194, 4.01753136566828914490718200427, 4.24678752068275589752388544941, 4.30641664752861511289395646882, 4.49094011885412783433562128268, 4.53720965940148724811634649469, 4.54856793771602274289158255176, 5.00673082250196025744122644956, 5.08969736985828003556101798591, 5.09677434211087661219120649232, 5.12015398653510732720568641796

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

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