Properties

Degree 4
Conductor $ 2^{4} \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

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(400\)    =    \(2^{4} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(9\)
character  :  induced by $\chi_{20} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 400,\ (\ :9/2, 9/2),\ 1)\)
\(L(5)\)  \(\approx\)  \(1.11465\)
\(L(\frac12)\)  \(\approx\)  \(1.11465\)
\(L(\frac{11}{2})\)   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

−16.51122131683281429736759222612, −16.21723031146093919389283279494, −15.40185298097005058775829723939, −14.67269881407497906909611779116, −13.91493092406230454736814601987, −12.97609545859416210930129435340, −12.13656968920087926388401352623, −11.78884887211767794359215900147, −11.14289241511026106393967605489, −10.92628573734094939785313602525, −9.639685230494375499020864739510, −8.930418937147735166878198111736, −7.79655775891205757910738772391, −6.89071308966658125369361898257, −5.99825631602642481469719988168, −5.68085486815625618824700954297, −4.10967824369615132705621681419, −3.76558598593066383210248116523, −1.32686442479705478019279561910, −0.65104259923725667941952006804, 0.65104259923725667941952006804, 1.32686442479705478019279561910, 3.76558598593066383210248116523, 4.10967824369615132705621681419, 5.68085486815625618824700954297, 5.99825631602642481469719988168, 6.89071308966658125369361898257, 7.79655775891205757910738772391, 8.930418937147735166878198111736, 9.639685230494375499020864739510, 10.92628573734094939785313602525, 11.14289241511026106393967605489, 11.78884887211767794359215900147, 12.13656968920087926388401352623, 12.97609545859416210930129435340, 13.91493092406230454736814601987, 14.67269881407497906909611779116, 15.40185298097005058775829723939, 16.21723031146093919389283279494, 16.51122131683281429736759222612

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