Properties

Degree 16
Conductor $ 2^{32} \cdot 3^{16} \cdot 7^{8} $
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  + 4·3-s + 4·5-s + 4·7-s + 13·9-s + 6·11-s − 3·13-s + 16·15-s − 16·17-s + 4·19-s + 16·21-s + 5·23-s + 11·25-s + 29·27-s + 29-s − 11·31-s + 24·33-s + 16·35-s + 54·37-s − 12·39-s + 2·41-s + 11·43-s + 52·45-s − 7·47-s + 6·49-s − 64·51-s − 8·53-s + 24·55-s + ⋯
L(s)  = 1  + 2.30·3-s + 1.78·5-s + 1.51·7-s + 13/3·9-s + 1.80·11-s − 0.832·13-s + 4.13·15-s − 3.88·17-s + 0.917·19-s + 3.49·21-s + 1.04·23-s + 11/5·25-s + 5.58·27-s + 0.185·29-s − 1.97·31-s + 4.17·33-s + 2.70·35-s + 8.87·37-s − 1.92·39-s + 0.312·41-s + 1.67·43-s + 7.75·45-s − 1.02·47-s + 6/7·49-s − 8.96·51-s − 1.09·53-s + 3.23·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{32} \cdot 3^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{32} \cdot 3^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(62.97632890\)
\(L(\frac12)\)  \(\approx\)  \(62.97632890\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 4 T + p T^{2} + 11 T^{3} - 32 T^{4} + 11 p T^{5} + p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
7 \( ( 1 - T + T^{2} )^{4} \)
good5 \( 1 - 4 T + p T^{2} + 18 T^{3} - 94 T^{4} + 232 T^{5} - 22 p T^{6} - 1011 T^{7} + 3826 T^{8} - 1011 p T^{9} - 22 p^{3} T^{10} + 232 p^{3} T^{11} - 94 p^{4} T^{12} + 18 p^{5} T^{13} + p^{7} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 - 6 T + T^{2} + 24 T^{3} + 49 T^{4} + 306 T^{5} - 2153 T^{6} + 1236 T^{7} + 7063 T^{8} + 1236 p T^{9} - 2153 p^{2} T^{10} + 306 p^{3} T^{11} + 49 p^{4} T^{12} + 24 p^{5} T^{13} + p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 + 3 T - 16 T^{2} + 3 p T^{3} + 337 T^{4} - 720 T^{5} + 2222 T^{6} + 906 p T^{7} - 3392 p T^{8} + 906 p^{2} T^{9} + 2222 p^{2} T^{10} - 720 p^{3} T^{11} + 337 p^{4} T^{12} + 3 p^{6} T^{13} - 16 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \)
17 \( ( 1 + 8 T + 35 T^{2} + 167 T^{3} + 886 T^{4} + 167 p T^{5} + 35 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 2 T + 55 T^{2} - 41 T^{3} + 1309 T^{4} - 41 p T^{5} + 55 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 5 T - 28 T^{2} + 177 T^{3} - 25 T^{4} + 494 T^{5} - 12137 T^{6} - 44331 T^{7} + 708646 T^{8} - 44331 p T^{9} - 12137 p^{2} T^{10} + 494 p^{3} T^{11} - 25 p^{4} T^{12} + 177 p^{5} T^{13} - 28 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - T - 49 T^{2} + 294 T^{3} + 1136 T^{4} - 11204 T^{5} + 49096 T^{6} + 227337 T^{7} - 2198075 T^{8} + 227337 p T^{9} + 49096 p^{2} T^{10} - 11204 p^{3} T^{11} + 1136 p^{4} T^{12} + 294 p^{5} T^{13} - 49 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 + 11 T - 39 T^{2} - 356 T^{3} + 5954 T^{4} + 25902 T^{5} - 215174 T^{6} - 3835 T^{7} + 11112081 T^{8} - 3835 p T^{9} - 215174 p^{2} T^{10} + 25902 p^{3} T^{11} + 5954 p^{4} T^{12} - 356 p^{5} T^{13} - 39 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \)
37 \( ( 1 - 27 T + 412 T^{2} - 4104 T^{3} + 29436 T^{4} - 4104 p T^{5} + 412 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 - 2 T - 37 T^{2} + 432 T^{3} - 841 T^{4} - 13042 T^{5} + 100489 T^{6} + 72294 T^{7} - 3776099 T^{8} + 72294 p T^{9} + 100489 p^{2} T^{10} - 13042 p^{3} T^{11} - 841 p^{4} T^{12} + 432 p^{5} T^{13} - 37 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 11 T - 27 T^{2} + 596 T^{3} + 140 T^{4} - 2688 T^{5} - 152864 T^{6} - 330797 T^{7} + 13009029 T^{8} - 330797 p T^{9} - 152864 p^{2} T^{10} - 2688 p^{3} T^{11} + 140 p^{4} T^{12} + 596 p^{5} T^{13} - 27 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 7 T - 61 T^{2} + 174 T^{3} + 5108 T^{4} - 21898 T^{5} - 62918 T^{6} + 808359 T^{7} - 1538879 T^{8} + 808359 p T^{9} - 62918 p^{2} T^{10} - 21898 p^{3} T^{11} + 5108 p^{4} T^{12} + 174 p^{5} T^{13} - 61 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
53 \( ( 1 + 4 T + 83 T^{2} - 197 T^{3} + 2722 T^{4} - 197 p T^{5} + 83 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 + 9 T - 119 T^{2} - 1116 T^{3} + 10336 T^{4} + 79866 T^{5} - 564434 T^{6} - 2176353 T^{7} + 30131725 T^{8} - 2176353 p T^{9} - 564434 p^{2} T^{10} + 79866 p^{3} T^{11} + 10336 p^{4} T^{12} - 1116 p^{5} T^{13} - 119 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 7 T - 144 T^{2} - 1315 T^{3} + 11783 T^{4} + 112692 T^{5} - 470051 T^{6} - 3262925 T^{7} + 21742764 T^{8} - 3262925 p T^{9} - 470051 p^{2} T^{10} + 112692 p^{3} T^{11} + 11783 p^{4} T^{12} - 1315 p^{5} T^{13} - 144 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 12 T - 139 T^{2} + 1338 T^{3} + 22729 T^{4} - 133506 T^{5} - 1954153 T^{6} + 2368722 T^{7} + 170018875 T^{8} + 2368722 p T^{9} - 1954153 p^{2} T^{10} - 133506 p^{3} T^{11} + 22729 p^{4} T^{12} + 1338 p^{5} T^{13} - 139 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
71 \( ( 1 - 12 T + 167 T^{2} - 591 T^{3} + 7587 T^{4} - 591 p T^{5} + 167 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 13 T + 232 T^{2} - 1504 T^{3} + 18820 T^{4} - 1504 p T^{5} + 232 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 - 22 T + 3 p T^{2} - 1616 T^{3} + 2882 T^{4} + 87342 T^{5} - 574370 T^{6} - 1050079 T^{7} + 29800944 T^{8} - 1050079 p T^{9} - 574370 p^{2} T^{10} + 87342 p^{3} T^{11} + 2882 p^{4} T^{12} - 1616 p^{5} T^{13} + 3 p^{7} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 - 6 T - 113 T^{2} + 1704 T^{3} + 3151 T^{4} - 166296 T^{5} + 1416277 T^{6} + 7918434 T^{7} - 172589093 T^{8} + 7918434 p T^{9} + 1416277 p^{2} T^{10} - 166296 p^{3} T^{11} + 3151 p^{4} T^{12} + 1704 p^{5} T^{13} - 113 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 + 14 T + 323 T^{2} + 2963 T^{3} + 41152 T^{4} + 2963 p T^{5} + 323 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 + T - 123 T^{2} - 856 T^{3} - 4138 T^{4} + 70242 T^{5} + 167026 T^{6} - 386291 T^{7} + 107246157 T^{8} - 386291 p T^{9} + 167026 p^{2} T^{10} + 70242 p^{3} T^{11} - 4138 p^{4} T^{12} - 856 p^{5} T^{13} - 123 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.31002218442586362228626244198, −4.20392260048891755972278103661, −4.16844378365029645867305905813, −3.71681788162767728136194258438, −3.69460458261305999086505274716, −3.68596881766531208926260529681, −3.56041807731883871826669307090, −3.45051427316684619735547221347, −3.14987641084811615120284024332, −3.00679650523693520568243175533, −2.61862361815010886389404078914, −2.57238193044226460596354933740, −2.51417421035978497324770397266, −2.38649886124281136422978392741, −2.34195032338292114553405492504, −2.32398024122800783458981058269, −2.08586393242414142155157691636, −1.98054941973034237064094665683, −1.65742367255365513695835255093, −1.55429264845187736571633896404, −1.12445364140883091083139610497, −1.11070930205010049140711775876, −1.07576780015636935545019452205, −0.977220995333953795482071952560, −0.40871027364755789465793851418, 0.40871027364755789465793851418, 0.977220995333953795482071952560, 1.07576780015636935545019452205, 1.11070930205010049140711775876, 1.12445364140883091083139610497, 1.55429264845187736571633896404, 1.65742367255365513695835255093, 1.98054941973034237064094665683, 2.08586393242414142155157691636, 2.32398024122800783458981058269, 2.34195032338292114553405492504, 2.38649886124281136422978392741, 2.51417421035978497324770397266, 2.57238193044226460596354933740, 2.61862361815010886389404078914, 3.00679650523693520568243175533, 3.14987641084811615120284024332, 3.45051427316684619735547221347, 3.56041807731883871826669307090, 3.68596881766531208926260529681, 3.69460458261305999086505274716, 3.71681788162767728136194258438, 4.16844378365029645867305905813, 4.20392260048891755972278103661, 4.31002218442586362228626244198

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

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