Properties

Degree 32
Conductor $ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} $
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  − 16·2-s + 128·4-s − 16·5-s − 672·8-s + 256·10-s + 8·11-s − 32·13-s + 2.54e3·16-s + 56·17-s − 2.04e3·20-s − 128·22-s + 24·23-s + 148·25-s + 512·26-s − 112·31-s − 7.10e3·32-s − 896·34-s − 152·37-s + 1.07e4·40-s + 1.02e3·44-s − 384·46-s + 80·47-s − 2.36e3·50-s − 4.09e3·52-s + 48·53-s − 128·55-s + 96·61-s + ⋯
L(s)  = 1  − 8·2-s + 32·4-s − 3.19·5-s − 84·8-s + 25.5·10-s + 8/11·11-s − 2.46·13-s + 159·16-s + 3.29·17-s − 102.·20-s − 5.81·22-s + 1.04·23-s + 5.91·25-s + 19.6·26-s − 3.61·31-s − 222·32-s − 26.3·34-s − 4.10·37-s + 268.·40-s + 23.2·44-s − 8.34·46-s + 1.70·47-s − 47.3·50-s − 78.7·52-s + 0.905·53-s − 2.32·55-s + 1.57·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{210} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.00500243\)
\(L(\frac12)\)  \(\approx\)  \(0.00500243\)
\(L(2)\)   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 32. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( ( 1 + p T + p T^{2} )^{8} \)
3 \( ( 1 + p^{2} T^{4} )^{4} \)
5 \( 1 + 16 T + 108 T^{2} + 48 p T^{3} - 1736 T^{4} - 15856 T^{5} - 26268 T^{6} + 70704 p T^{7} + 117534 p^{2} T^{8} + 70704 p^{3} T^{9} - 26268 p^{4} T^{10} - 15856 p^{6} T^{11} - 1736 p^{8} T^{12} + 48 p^{11} T^{13} + 108 p^{12} T^{14} + 16 p^{14} T^{15} + p^{16} T^{16} \)
7 \( ( 1 + p^{2} T^{4} )^{4} \)
good11 \( ( 1 - 4 T + 428 T^{2} - 3020 T^{3} + 112132 T^{4} - 750180 T^{5} + 21255892 T^{6} - 12021156 p T^{7} + 2897491254 T^{8} - 12021156 p^{3} T^{9} + 21255892 p^{4} T^{10} - 750180 p^{6} T^{11} + 112132 p^{8} T^{12} - 3020 p^{10} T^{13} + 428 p^{12} T^{14} - 4 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
13 \( 1 + 32 T + 512 T^{2} + 12624 T^{3} + 210400 T^{4} + 1218608 T^{5} + 10953344 T^{6} - 12361888 T^{7} - 9595251012 T^{8} - 144909422048 T^{9} - 1442470310272 T^{10} - 30698489798640 T^{11} - 15354999406944 p T^{12} + 2586969858854768 T^{13} + 29532641854346496 T^{14} + 799237423444942944 T^{15} + 19007129532539001606 T^{16} + 799237423444942944 p^{2} T^{17} + 29532641854346496 p^{4} T^{18} + 2586969858854768 p^{6} T^{19} - 15354999406944 p^{9} T^{20} - 30698489798640 p^{10} T^{21} - 1442470310272 p^{12} T^{22} - 144909422048 p^{14} T^{23} - 9595251012 p^{16} T^{24} - 12361888 p^{18} T^{25} + 10953344 p^{20} T^{26} + 1218608 p^{22} T^{27} + 210400 p^{24} T^{28} + 12624 p^{26} T^{29} + 512 p^{28} T^{30} + 32 p^{30} T^{31} + p^{32} T^{32} \)
17 \( 1 - 56 T + 1568 T^{2} - 29832 T^{3} + 267808 T^{4} + 199880 p T^{5} - 165234592 T^{6} + 3465012472 T^{7} - 48572233284 T^{8} + 491667950696 T^{9} - 4716183819232 T^{10} + 39445096533336 T^{11} + 280624802792928 T^{12} - 26208629063564312 T^{13} + 761979769352889696 T^{14} - 15987258252632453928 T^{15} + \)\(28\!\cdots\!74\)\( T^{16} - 15987258252632453928 p^{2} T^{17} + 761979769352889696 p^{4} T^{18} - 26208629063564312 p^{6} T^{19} + 280624802792928 p^{8} T^{20} + 39445096533336 p^{10} T^{21} - 4716183819232 p^{12} T^{22} + 491667950696 p^{14} T^{23} - 48572233284 p^{16} T^{24} + 3465012472 p^{18} T^{25} - 165234592 p^{20} T^{26} + 199880 p^{23} T^{27} + 267808 p^{24} T^{28} - 29832 p^{26} T^{29} + 1568 p^{28} T^{30} - 56 p^{30} T^{31} + p^{32} T^{32} \)
19 \( 1 - 2056 T^{2} + 2230944 T^{4} - 1603740248 T^{6} + 43114975316 p T^{8} - 297045269538504 T^{10} + 69987022118810464 T^{12} - 7175141044809308632 T^{14} - \)\(43\!\cdots\!46\)\( T^{16} - 7175141044809308632 p^{4} T^{18} + 69987022118810464 p^{8} T^{20} - 297045269538504 p^{12} T^{22} + 43114975316 p^{17} T^{24} - 1603740248 p^{20} T^{26} + 2230944 p^{24} T^{28} - 2056 p^{28} T^{30} + p^{32} T^{32} \)
23 \( 1 - 24 T + 288 T^{2} - 17192 T^{3} + 221472 T^{4} - 2391768 T^{5} + 141400928 T^{6} - 3172207336 T^{7} + 137674628156 T^{8} - 274022502200 T^{9} - 6993233578720 T^{10} - 103772124341000 T^{11} - 60508914798085408 T^{12} + 933592691352233992 T^{13} - 15427273581773601184 T^{14} + \)\(57\!\cdots\!36\)\( T^{15} - \)\(64\!\cdots\!26\)\( T^{16} + \)\(57\!\cdots\!36\)\( p^{2} T^{17} - 15427273581773601184 p^{4} T^{18} + 933592691352233992 p^{6} T^{19} - 60508914798085408 p^{8} T^{20} - 103772124341000 p^{10} T^{21} - 6993233578720 p^{12} T^{22} - 274022502200 p^{14} T^{23} + 137674628156 p^{16} T^{24} - 3172207336 p^{18} T^{25} + 141400928 p^{20} T^{26} - 2391768 p^{22} T^{27} + 221472 p^{24} T^{28} - 17192 p^{26} T^{29} + 288 p^{28} T^{30} - 24 p^{30} T^{31} + p^{32} T^{32} \)
29 \( 1 - 4032 T^{2} + 8105976 T^{4} - 11130637632 T^{6} + 11901856158364 T^{8} - 11208853722734016 T^{10} + 10516466734892486856 T^{12} - \)\(99\!\cdots\!40\)\( T^{14} + \)\(87\!\cdots\!46\)\( T^{16} - \)\(99\!\cdots\!40\)\( p^{4} T^{18} + 10516466734892486856 p^{8} T^{20} - 11208853722734016 p^{12} T^{22} + 11901856158364 p^{16} T^{24} - 11130637632 p^{20} T^{26} + 8105976 p^{24} T^{28} - 4032 p^{28} T^{30} + p^{32} T^{32} \)
31 \( ( 1 + 56 T + 4484 T^{2} + 186776 T^{3} + 8992264 T^{4} + 304202984 T^{5} + 11800345036 T^{6} + 348556649544 T^{7} + 12266333612430 T^{8} + 348556649544 p^{2} T^{9} + 11800345036 p^{4} T^{10} + 304202984 p^{6} T^{11} + 8992264 p^{8} T^{12} + 186776 p^{10} T^{13} + 4484 p^{12} T^{14} + 56 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
37 \( 1 + 152 T + 11552 T^{2} + 598344 T^{3} + 25962168 T^{4} + 1167158904 T^{5} + 56500959840 T^{6} + 2639490056104 T^{7} + 111109511071516 T^{8} + 4242418231122872 T^{9} + 154876079427922976 T^{10} + 5762441784032775208 T^{11} + \)\(22\!\cdots\!72\)\( T^{12} + \)\(89\!\cdots\!08\)\( T^{13} + \)\(32\!\cdots\!12\)\( T^{14} + \)\(10\!\cdots\!40\)\( T^{15} + \)\(37\!\cdots\!46\)\( T^{16} + \)\(10\!\cdots\!40\)\( p^{2} T^{17} + \)\(32\!\cdots\!12\)\( p^{4} T^{18} + \)\(89\!\cdots\!08\)\( p^{6} T^{19} + \)\(22\!\cdots\!72\)\( p^{8} T^{20} + 5762441784032775208 p^{10} T^{21} + 154876079427922976 p^{12} T^{22} + 4242418231122872 p^{14} T^{23} + 111109511071516 p^{16} T^{24} + 2639490056104 p^{18} T^{25} + 56500959840 p^{20} T^{26} + 1167158904 p^{22} T^{27} + 25962168 p^{24} T^{28} + 598344 p^{26} T^{29} + 11552 p^{28} T^{30} + 152 p^{30} T^{31} + p^{32} T^{32} \)
41 \( ( 1 + 8252 T^{2} - 19936 T^{3} + 34068728 T^{4} - 127251488 T^{5} + 92291784532 T^{6} - 367862933760 T^{7} + 180432835211182 T^{8} - 367862933760 p^{2} T^{9} + 92291784532 p^{4} T^{10} - 127251488 p^{6} T^{11} + 34068728 p^{8} T^{12} - 19936 p^{10} T^{13} + 8252 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
43 \( 1 - 158592 T^{3} + 2725896 T^{4} - 61004160 T^{5} + 12575711232 T^{6} - 618250768128 T^{7} + 18017190018460 T^{8} - 968512496926464 T^{9} + 65630298643144704 T^{10} - 3092793950726971008 T^{11} + \)\(10\!\cdots\!80\)\( T^{12} - \)\(37\!\cdots\!52\)\( T^{13} + \)\(33\!\cdots\!64\)\( T^{14} - \)\(12\!\cdots\!48\)\( T^{15} + \)\(20\!\cdots\!26\)\( T^{16} - \)\(12\!\cdots\!48\)\( p^{2} T^{17} + \)\(33\!\cdots\!64\)\( p^{4} T^{18} - \)\(37\!\cdots\!52\)\( p^{6} T^{19} + \)\(10\!\cdots\!80\)\( p^{8} T^{20} - 3092793950726971008 p^{10} T^{21} + 65630298643144704 p^{12} T^{22} - 968512496926464 p^{14} T^{23} + 18017190018460 p^{16} T^{24} - 618250768128 p^{18} T^{25} + 12575711232 p^{20} T^{26} - 61004160 p^{22} T^{27} + 2725896 p^{24} T^{28} - 158592 p^{26} T^{29} + p^{32} T^{32} \)
47 \( 1 - 80 T + 3200 T^{2} - 185424 T^{3} + 10961416 T^{4} - 416599088 T^{5} + 15442425728 T^{6} - 918151881776 T^{7} + 40326873674268 T^{8} - 452413353633424 T^{9} - 5098985333045632 T^{10} + 1563817770500267376 T^{11} - \)\(31\!\cdots\!76\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{13} - \)\(66\!\cdots\!16\)\( T^{14} + \)\(36\!\cdots\!52\)\( T^{15} - \)\(19\!\cdots\!38\)\( T^{16} + \)\(36\!\cdots\!52\)\( p^{2} T^{17} - \)\(66\!\cdots\!16\)\( p^{4} T^{18} + \)\(17\!\cdots\!60\)\( p^{6} T^{19} - \)\(31\!\cdots\!76\)\( p^{8} T^{20} + 1563817770500267376 p^{10} T^{21} - 5098985333045632 p^{12} T^{22} - 452413353633424 p^{14} T^{23} + 40326873674268 p^{16} T^{24} - 918151881776 p^{18} T^{25} + 15442425728 p^{20} T^{26} - 416599088 p^{22} T^{27} + 10961416 p^{24} T^{28} - 185424 p^{26} T^{29} + 3200 p^{28} T^{30} - 80 p^{30} T^{31} + p^{32} T^{32} \)
53 \( 1 - 48 T + 1152 T^{2} - 172480 T^{3} + 14460640 T^{4} - 33746752 T^{5} - 164137984 T^{6} + 7272556080 T^{7} - 144052676687940 T^{8} + 6754246786116880 T^{9} - 156566570729557248 T^{10} + 21067810577807322688 T^{11} - \)\(28\!\cdots\!60\)\( T^{12} - \)\(50\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!80\)\( T^{14} - \)\(15\!\cdots\!88\)\( T^{15} + \)\(20\!\cdots\!78\)\( T^{16} - \)\(15\!\cdots\!88\)\( p^{2} T^{17} + \)\(11\!\cdots\!80\)\( p^{4} T^{18} - \)\(50\!\cdots\!00\)\( p^{6} T^{19} - \)\(28\!\cdots\!60\)\( p^{8} T^{20} + 21067810577807322688 p^{10} T^{21} - 156566570729557248 p^{12} T^{22} + 6754246786116880 p^{14} T^{23} - 144052676687940 p^{16} T^{24} + 7272556080 p^{18} T^{25} - 164137984 p^{20} T^{26} - 33746752 p^{22} T^{27} + 14460640 p^{24} T^{28} - 172480 p^{26} T^{29} + 1152 p^{28} T^{30} - 48 p^{30} T^{31} + p^{32} T^{32} \)
59 \( 1 - 14320 T^{2} + 157767608 T^{4} - 1244890527696 T^{6} + 8207935216533276 T^{8} - 45520264315988706288 T^{10} + \)\(21\!\cdots\!36\)\( T^{12} - \)\(91\!\cdots\!96\)\( T^{14} + \)\(33\!\cdots\!38\)\( T^{16} - \)\(91\!\cdots\!96\)\( p^{4} T^{18} + \)\(21\!\cdots\!36\)\( p^{8} T^{20} - 45520264315988706288 p^{12} T^{22} + 8207935216533276 p^{16} T^{24} - 1244890527696 p^{20} T^{26} + 157767608 p^{24} T^{28} - 14320 p^{28} T^{30} + p^{32} T^{32} \)
61 \( ( 1 - 48 T + 22640 T^{2} - 673936 T^{3} + 215932124 T^{4} - 3175993520 T^{5} + 1217381291920 T^{6} - 6110146782096 T^{7} + 5029652273881990 T^{8} - 6110146782096 p^{2} T^{9} + 1217381291920 p^{4} T^{10} - 3175993520 p^{6} T^{11} + 215932124 p^{8} T^{12} - 673936 p^{10} T^{13} + 22640 p^{12} T^{14} - 48 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
67 \( 1 + 80 T + 3200 T^{2} - 9104 p T^{3} - 25803832 T^{4} + 2674348464 T^{5} + 482550620032 T^{6} + 12521071726512 T^{7} - 1529742891382500 T^{8} - 116256625005179504 T^{9} + 3984919157383710336 T^{10} + \)\(89\!\cdots\!92\)\( T^{11} + \)\(25\!\cdots\!88\)\( T^{12} - \)\(21\!\cdots\!76\)\( T^{13} - \)\(18\!\cdots\!68\)\( T^{14} + \)\(71\!\cdots\!08\)\( T^{15} + \)\(10\!\cdots\!46\)\( T^{16} + \)\(71\!\cdots\!08\)\( p^{2} T^{17} - \)\(18\!\cdots\!68\)\( p^{4} T^{18} - \)\(21\!\cdots\!76\)\( p^{6} T^{19} + \)\(25\!\cdots\!88\)\( p^{8} T^{20} + \)\(89\!\cdots\!92\)\( p^{10} T^{21} + 3984919157383710336 p^{12} T^{22} - 116256625005179504 p^{14} T^{23} - 1529742891382500 p^{16} T^{24} + 12521071726512 p^{18} T^{25} + 482550620032 p^{20} T^{26} + 2674348464 p^{22} T^{27} - 25803832 p^{24} T^{28} - 9104 p^{27} T^{29} + 3200 p^{28} T^{30} + 80 p^{30} T^{31} + p^{32} T^{32} \)
71 \( ( 1 - 268 T + 44204 T^{2} - 5071044 T^{3} + 471192708 T^{4} - 37316092140 T^{5} + 2755016710036 T^{6} - 195026045020868 T^{7} + 13901563235868662 T^{8} - 195026045020868 p^{2} T^{9} + 2755016710036 p^{4} T^{10} - 37316092140 p^{6} T^{11} + 471192708 p^{8} T^{12} - 5071044 p^{10} T^{13} + 44204 p^{12} T^{14} - 268 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
73 \( 1 - 638960 T^{3} - 86919200 T^{4} - 369113360 T^{5} + 204134940800 T^{6} + 33110468899200 T^{7} + 5206311620706876 T^{8} + 26720496193596800 T^{9} - 3344897125184227200 T^{10} - \)\(13\!\cdots\!20\)\( T^{11} - \)\(22\!\cdots\!00\)\( T^{12} - \)\(53\!\cdots\!20\)\( T^{13} + \)\(75\!\cdots\!00\)\( T^{14} + \)\(36\!\cdots\!00\)\( T^{15} + \)\(78\!\cdots\!66\)\( T^{16} + \)\(36\!\cdots\!00\)\( p^{2} T^{17} + \)\(75\!\cdots\!00\)\( p^{4} T^{18} - \)\(53\!\cdots\!20\)\( p^{6} T^{19} - \)\(22\!\cdots\!00\)\( p^{8} T^{20} - \)\(13\!\cdots\!20\)\( p^{10} T^{21} - 3344897125184227200 p^{12} T^{22} + 26720496193596800 p^{14} T^{23} + 5206311620706876 p^{16} T^{24} + 33110468899200 p^{18} T^{25} + 204134940800 p^{20} T^{26} - 369113360 p^{22} T^{27} - 86919200 p^{24} T^{28} - 638960 p^{26} T^{29} + p^{32} T^{32} \)
79 \( 1 - 52208 T^{2} + 1391680568 T^{4} - 25119513250000 T^{6} + 343967562165562012 T^{8} - \)\(37\!\cdots\!72\)\( T^{10} + \)\(34\!\cdots\!24\)\( T^{12} - \)\(27\!\cdots\!40\)\( T^{14} + \)\(18\!\cdots\!30\)\( T^{16} - \)\(27\!\cdots\!40\)\( p^{4} T^{18} + \)\(34\!\cdots\!24\)\( p^{8} T^{20} - \)\(37\!\cdots\!72\)\( p^{12} T^{22} + 343967562165562012 p^{16} T^{24} - 25119513250000 p^{20} T^{26} + 1391680568 p^{24} T^{28} - 52208 p^{28} T^{30} + p^{32} T^{32} \)
83 \( 1 + 256 T + 32768 T^{2} + 2489472 T^{3} + 78882184 T^{4} + 642592896 T^{5} + 678427795456 T^{6} + 150782876556800 T^{7} + 17212084280490396 T^{8} + 1275723698299503616 T^{9} + 84640034952293916672 T^{10} + \)\(62\!\cdots\!76\)\( T^{11} + \)\(45\!\cdots\!48\)\( T^{12} + \)\(40\!\cdots\!04\)\( T^{13} + \)\(47\!\cdots\!44\)\( T^{14} + \)\(59\!\cdots\!24\)\( T^{15} + \)\(59\!\cdots\!02\)\( T^{16} + \)\(59\!\cdots\!24\)\( p^{2} T^{17} + \)\(47\!\cdots\!44\)\( p^{4} T^{18} + \)\(40\!\cdots\!04\)\( p^{6} T^{19} + \)\(45\!\cdots\!48\)\( p^{8} T^{20} + \)\(62\!\cdots\!76\)\( p^{10} T^{21} + 84640034952293916672 p^{12} T^{22} + 1275723698299503616 p^{14} T^{23} + 17212084280490396 p^{16} T^{24} + 150782876556800 p^{18} T^{25} + 678427795456 p^{20} T^{26} + 642592896 p^{22} T^{27} + 78882184 p^{24} T^{28} + 2489472 p^{26} T^{29} + 32768 p^{28} T^{30} + 256 p^{30} T^{31} + p^{32} T^{32} \)
89 \( 1 - 60568 T^{2} + 1632136448 T^{4} - 25689569431240 T^{6} + 260052896713761532 T^{8} - \)\(17\!\cdots\!32\)\( T^{10} + \)\(70\!\cdots\!24\)\( T^{12} - \)\(68\!\cdots\!20\)\( T^{14} - \)\(92\!\cdots\!90\)\( T^{16} - \)\(68\!\cdots\!20\)\( p^{4} T^{18} + \)\(70\!\cdots\!24\)\( p^{8} T^{20} - \)\(17\!\cdots\!32\)\( p^{12} T^{22} + 260052896713761532 p^{16} T^{24} - 25689569431240 p^{20} T^{26} + 1632136448 p^{24} T^{28} - 60568 p^{28} T^{30} + p^{32} T^{32} \)
97 \( 1 - 688 T + 236672 T^{2} - 56991072 T^{3} + 10854023776 T^{4} - 1693042723360 T^{5} + 219961022412800 T^{6} - 24020408470599760 T^{7} + 2158083026478171708 T^{8} - \)\(14\!\cdots\!44\)\( T^{9} + \)\(52\!\cdots\!16\)\( T^{10} + \)\(44\!\cdots\!04\)\( T^{11} - \)\(12\!\cdots\!36\)\( T^{12} + \)\(17\!\cdots\!52\)\( T^{13} - \)\(20\!\cdots\!48\)\( T^{14} + \)\(21\!\cdots\!08\)\( T^{15} - \)\(20\!\cdots\!18\)\( T^{16} + \)\(21\!\cdots\!08\)\( p^{2} T^{17} - \)\(20\!\cdots\!48\)\( p^{4} T^{18} + \)\(17\!\cdots\!52\)\( p^{6} T^{19} - \)\(12\!\cdots\!36\)\( p^{8} T^{20} + \)\(44\!\cdots\!04\)\( p^{10} T^{21} + \)\(52\!\cdots\!16\)\( p^{12} T^{22} - \)\(14\!\cdots\!44\)\( p^{14} T^{23} + 2158083026478171708 p^{16} T^{24} - 24020408470599760 p^{18} T^{25} + 219961022412800 p^{20} T^{26} - 1693042723360 p^{22} T^{27} + 10854023776 p^{24} T^{28} - 56991072 p^{26} T^{29} + 236672 p^{28} T^{30} - 688 p^{30} T^{31} + p^{32} T^{32} \)
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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

−3.22072595984949487759906627580, −3.09618263054485291305880923166, −2.94037550267987467208196855000, −2.87179214096906934885447081707, −2.76121276109149525847600862461, −2.55350167610511474621094216171, −2.47425528455941239649156362507, −2.43666906040870691427659470823, −2.38338886041956038850265206394, −2.21309747250630967393251974379, −2.08163893887610663699410363479, −1.94338449547543272604344697677, −1.73302924180320869885019274010, −1.71009064446271767592586966835, −1.39950008238396848672532526061, −1.31220516719191800821502814560, −1.29110722981586088033091311054, −1.25736055090861230815544012856, −1.18782990679096045559197901198, −0.77416417535641666486886929283, −0.61852412234657013608966523889, −0.49606625727679552315257615614, −0.47496334754704197397559962942, −0.26025988267906733507163642398, −0.11041933225371415237369802690, 0.11041933225371415237369802690, 0.26025988267906733507163642398, 0.47496334754704197397559962942, 0.49606625727679552315257615614, 0.61852412234657013608966523889, 0.77416417535641666486886929283, 1.18782990679096045559197901198, 1.25736055090861230815544012856, 1.29110722981586088033091311054, 1.31220516719191800821502814560, 1.39950008238396848672532526061, 1.71009064446271767592586966835, 1.73302924180320869885019274010, 1.94338449547543272604344697677, 2.08163893887610663699410363479, 2.21309747250630967393251974379, 2.38338886041956038850265206394, 2.43666906040870691427659470823, 2.47425528455941239649156362507, 2.55350167610511474621094216171, 2.76121276109149525847600862461, 2.87179214096906934885447081707, 2.94037550267987467208196855000, 3.09618263054485291305880923166, 3.22072595984949487759906627580

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

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