Properties

Degree 4
Conductor $ 2^{8} \cdot 5^{2} $
Sign $1$
Motivic weight 9
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 260·3-s − 1.25e3·5-s + 380·7-s + 3.15e4·9-s − 1.02e5·11-s + 1.79e5·13-s − 3.25e5·15-s + 3.16e5·17-s − 1.37e5·19-s + 9.88e4·21-s + 6.65e5·23-s + 1.17e6·25-s + 3.95e6·27-s − 6.89e6·29-s − 2.91e5·31-s − 2.67e7·33-s − 4.75e5·35-s + 1.12e7·37-s + 4.65e7·39-s + 2.97e7·41-s + 1.17e7·43-s − 3.94e7·45-s − 6.24e7·47-s + 1.56e7·49-s + 8.21e7·51-s + 9.41e6·53-s + 1.28e8·55-s + ⋯
L(s)  = 1  + 1.85·3-s − 0.894·5-s + 0.0598·7-s + 1.60·9-s − 2.11·11-s + 1.73·13-s − 1.65·15-s + 0.917·17-s − 0.241·19-s + 0.110·21-s + 0.495·23-s + 3/5·25-s + 1.43·27-s − 1.80·29-s − 0.0567·31-s − 3.92·33-s − 0.0535·35-s + 0.987·37-s + 3.22·39-s + 1.64·41-s + 0.522·43-s − 1.43·45-s − 1.86·47-s + 0.388·49-s + 1.70·51-s + 0.163·53-s + 1.89·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(6400\)    =    \(2^{8} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(9\)
character  :  induced by $\chi_{80} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 6400,\ (\ :9/2, 9/2),\ 1)\)
\(L(5)\)  \(\approx\)  \(4.253554163\)
\(L(\frac12)\)  \(\approx\)  \(4.253554163\)
\(L(\frac{11}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_1$ \( ( 1 + p^{4} T )^{2} \)
good3$D_{4}$ \( 1 - 260 T + 12014 p T^{2} - 260 p^{9} T^{3} + p^{18} T^{4} \)
7$D_{4}$ \( 1 - 380 T - 2220450 p T^{2} - 380 p^{9} T^{3} + p^{18} T^{4} \)
11$D_{4}$ \( 1 + 102720 T + 7335543382 T^{2} + 102720 p^{9} T^{3} + p^{18} T^{4} \)
13$D_{4}$ \( 1 - 1060 p^{2} T + 22798610142 T^{2} - 1060 p^{11} T^{3} + p^{18} T^{4} \)
17$D_{4}$ \( 1 - 316020 T + 259693705798 T^{2} - 316020 p^{9} T^{3} + p^{18} T^{4} \)
19$D_{4}$ \( 1 + 137272 T + 111610161654 T^{2} + 137272 p^{9} T^{3} + p^{18} T^{4} \)
23$D_{4}$ \( 1 - 665460 T + 2886450615250 T^{2} - 665460 p^{9} T^{3} + p^{18} T^{4} \)
29$D_{4}$ \( 1 + 6893748 T + 40195999658014 T^{2} + 6893748 p^{9} T^{3} + p^{18} T^{4} \)
31$D_{4}$ \( 1 + 291832 T + 38964935800398 T^{2} + 291832 p^{9} T^{3} + p^{18} T^{4} \)
37$D_{4}$ \( 1 - 11261380 T + 218879982937230 T^{2} - 11261380 p^{9} T^{3} + p^{18} T^{4} \)
41$D_{4}$ \( 1 - 29773452 T + 771012402449398 T^{2} - 29773452 p^{9} T^{3} + p^{18} T^{4} \)
43$D_{4}$ \( 1 - 11708180 T + 838769843899386 T^{2} - 11708180 p^{9} T^{3} + p^{18} T^{4} \)
47$D_{4}$ \( 1 + 62493300 T + 3177958884734338 T^{2} + 62493300 p^{9} T^{3} + p^{18} T^{4} \)
53$D_{4}$ \( 1 - 9417780 T + 5708185761526990 T^{2} - 9417780 p^{9} T^{3} + p^{18} T^{4} \)
59$D_{4}$ \( 1 - 92930856 T + 16656477955483462 T^{2} - 92930856 p^{9} T^{3} + p^{18} T^{4} \)
61$D_{4}$ \( 1 - 195673924 T + 22434263296171326 T^{2} - 195673924 p^{9} T^{3} + p^{18} T^{4} \)
67$D_{4}$ \( 1 - 219767420 T + 65652945987990090 T^{2} - 219767420 p^{9} T^{3} + p^{18} T^{4} \)
71$D_{4}$ \( 1 + 311207016 T + 76405636625293726 T^{2} + 311207016 p^{9} T^{3} + p^{18} T^{4} \)
73$D_{4}$ \( 1 + 99224060 T + 35402447061205782 T^{2} + 99224060 p^{9} T^{3} + p^{18} T^{4} \)
79$D_{4}$ \( 1 + 542261776 T + 313115996157615582 T^{2} + 542261776 p^{9} T^{3} + p^{18} T^{4} \)
83$D_{4}$ \( 1 - 1256915700 T + 768086791626261130 T^{2} - 1256915700 p^{9} T^{3} + p^{18} T^{4} \)
89$D_{4}$ \( 1 + 462291852 T + 159603168035249494 T^{2} + 462291852 p^{9} T^{3} + p^{18} T^{4} \)
97$D_{4}$ \( 1 - 1671716740 T + 2048690578856969670 T^{2} - 1671716740 p^{9} T^{3} + p^{18} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.84068827378847719978103644725, −12.78788106885807182529047581062, −11.40411922665608263512937179803, −11.30396273404985612526059191294, −10.46704514461595522720153607517, −10.08833766192098982628778700527, −9.076417438042029524733595193705, −8.887797492315292043178587465270, −8.084978094470079456474824737921, −7.85556157854428215172305336865, −7.57444298606610031088207856214, −6.55287221090046689569192113552, −5.61689747604127660022986325720, −4.99823418617938555890498141799, −3.77266775031600139948349671929, −3.75188220792903252954679008041, −2.74184717270147313341001694869, −2.51091472761180748878431757729, −1.36993850432270231033993946694, −0.53174934548957606983398735291, 0.53174934548957606983398735291, 1.36993850432270231033993946694, 2.51091472761180748878431757729, 2.74184717270147313341001694869, 3.75188220792903252954679008041, 3.77266775031600139948349671929, 4.99823418617938555890498141799, 5.61689747604127660022986325720, 6.55287221090046689569192113552, 7.57444298606610031088207856214, 7.85556157854428215172305336865, 8.084978094470079456474824737921, 8.887797492315292043178587465270, 9.076417438042029524733595193705, 10.08833766192098982628778700527, 10.46704514461595522720153607517, 11.30396273404985612526059191294, 11.40411922665608263512937179803, 12.78788106885807182529047581062, 12.84068827378847719978103644725

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