Properties

Degree 4
Conductor $ 2^{4} \cdot 5^{2} $
Sign $1$
Motivic weight 11
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 220·3-s + 6.25e3·5-s + 3.34e3·7-s − 1.31e5·9-s + 2.98e5·11-s + 1.20e6·13-s + 1.37e6·15-s + 9.05e6·17-s + 1.94e7·19-s + 7.34e5·21-s + 5.59e7·23-s + 2.92e7·25-s − 2.93e7·27-s + 4.18e7·29-s + 6.22e7·31-s + 6.56e7·33-s + 2.08e7·35-s − 7.29e8·37-s + 2.66e8·39-s − 6.08e8·41-s − 1.12e9·43-s − 8.19e8·45-s − 1.65e9·47-s − 1.64e9·49-s + 1.99e9·51-s + 5.72e9·53-s + 1.86e9·55-s + ⋯
L(s)  = 1  + 0.522·3-s + 0.894·5-s + 0.0751·7-s − 0.739·9-s + 0.558·11-s + 0.903·13-s + 0.467·15-s + 1.54·17-s + 1.80·19-s + 0.0392·21-s + 1.81·23-s + 3/5·25-s − 0.393·27-s + 0.378·29-s + 0.390·31-s + 0.291·33-s + 0.0671·35-s − 1.72·37-s + 0.472·39-s − 0.820·41-s − 1.17·43-s − 0.661·45-s − 1.05·47-s − 0.831·49-s + 0.808·51-s + 1.88·53-s + 0.499·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(400\)    =    \(2^{4} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(11\)
character  :  induced by $\chi_{20} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 400,\ (\ :11/2, 11/2),\ 1)$
$L(6)$  $\approx$  $4.57652$
$L(\frac12)$  $\approx$  $4.57652$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
5$C_1$ \( ( 1 - p^{5} T )^{2} \)
good3$D_{4}$ \( 1 - 220 T + 19942 p^{2} T^{2} - 220 p^{11} T^{3} + p^{22} T^{4} \)
7$D_{4}$ \( 1 - 3340 T + 236350050 p T^{2} - 3340 p^{11} T^{3} + p^{22} T^{4} \)
11$D_{4}$ \( 1 - 27120 p T + 564442123222 T^{2} - 27120 p^{12} T^{3} + p^{22} T^{4} \)
13$D_{4}$ \( 1 - 1209820 T + 2088490732398 T^{2} - 1209820 p^{11} T^{3} + p^{22} T^{4} \)
17$D_{4}$ \( 1 - 9056340 T + 84322733142022 T^{2} - 9056340 p^{11} T^{3} + p^{22} T^{4} \)
19$D_{4}$ \( 1 - 19439368 T + 284381272377894 T^{2} - 19439368 p^{11} T^{3} + p^{22} T^{4} \)
23$D_{4}$ \( 1 - 55926420 T + 116503185471650 p T^{2} - 55926420 p^{11} T^{3} + p^{22} T^{4} \)
29$D_{4}$ \( 1 - 41841708 T + 3630990695891374 T^{2} - 41841708 p^{11} T^{3} + p^{22} T^{4} \)
31$D_{4}$ \( 1 - 62230792 T - 13999135099203522 T^{2} - 62230792 p^{11} T^{3} + p^{22} T^{4} \)
37$D_{4}$ \( 1 + 729235940 T + 388817795213444670 T^{2} + 729235940 p^{11} T^{3} + p^{22} T^{4} \)
41$D_{4}$ \( 1 + 608419068 T + 1119141666750688438 T^{2} + 608419068 p^{11} T^{3} + p^{22} T^{4} \)
43$D_{4}$ \( 1 + 1129440740 T + 1605906671305010214 T^{2} + 1129440740 p^{11} T^{3} + p^{22} T^{4} \)
47$D_{4}$ \( 1 + 1653072900 T + 3734132615108598142 T^{2} + 1653072900 p^{11} T^{3} + p^{22} T^{4} \)
53$D_{4}$ \( 1 - 5724887340 T + 23865188747531536510 T^{2} - 5724887340 p^{11} T^{3} + p^{22} T^{4} \)
59$D_{4}$ \( 1 - 3756433896 T + 50973876172267502422 T^{2} - 3756433896 p^{11} T^{3} + p^{22} T^{4} \)
61$D_{4}$ \( 1 - 4923703564 T + 12712509351121558446 T^{2} - 4923703564 p^{11} T^{3} + p^{22} T^{4} \)
67$D_{4}$ \( 1 + 5843244140 T + \)\(24\!\cdots\!10\)\( T^{2} + 5843244140 p^{11} T^{3} + p^{22} T^{4} \)
71$D_{4}$ \( 1 + 43352162664 T + \)\(91\!\cdots\!66\)\( T^{2} + 43352162664 p^{11} T^{3} + p^{22} T^{4} \)
73$D_{4}$ \( 1 - 33647099620 T + \)\(88\!\cdots\!78\)\( T^{2} - 33647099620 p^{11} T^{3} + p^{22} T^{4} \)
79$D_{4}$ \( 1 + 54799425296 T + \)\(19\!\cdots\!62\)\( T^{2} + 54799425296 p^{11} T^{3} + p^{22} T^{4} \)
83$D_{4}$ \( 1 + 9303032100 T + \)\(25\!\cdots\!70\)\( T^{2} + 9303032100 p^{11} T^{3} + p^{22} T^{4} \)
89$D_{4}$ \( 1 - 3688968372 T + \)\(27\!\cdots\!74\)\( T^{2} - 3688968372 p^{11} T^{3} + p^{22} T^{4} \)
97$D_{4}$ \( 1 - 65892157780 T + \)\(13\!\cdots\!30\)\( T^{2} - 65892157780 p^{11} T^{3} + p^{22} T^{4} \)
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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

−16.03458385884730938452451840670, −15.25411397526198273628201349383, −14.34170056291467338563775045040, −14.26538754392787211991185845183, −13.47597716219513099343278854800, −12.95910073158939843814038225275, −11.68490758417089141112057801834, −11.64173366638452906321561310187, −10.30189389876211047645969176013, −9.905910460617328377855063822446, −8.846390771794035425364759206308, −8.622021605888535278546947734776, −7.44814632292907492963114435548, −6.62050569531529265957826017842, −5.58859370301800126600593293055, −5.08692856252929801595846713850, −3.34581210425957557176286113218, −3.08193399406185628290849167281, −1.58922700445621024259624905057, −0.931129079255647287645324693411, 0.931129079255647287645324693411, 1.58922700445621024259624905057, 3.08193399406185628290849167281, 3.34581210425957557176286113218, 5.08692856252929801595846713850, 5.58859370301800126600593293055, 6.62050569531529265957826017842, 7.44814632292907492963114435548, 8.622021605888535278546947734776, 8.846390771794035425364759206308, 9.905910460617328377855063822446, 10.30189389876211047645969176013, 11.64173366638452906321561310187, 11.68490758417089141112057801834, 12.95910073158939843814038225275, 13.47597716219513099343278854800, 14.26538754392787211991185845183, 14.34170056291467338563775045040, 15.25411397526198273628201349383, 16.03458385884730938452451840670

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