Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 20·2-s − 220·3-s + 1.64e3·4-s − 6.25e3·5-s + 4.40e3·6-s + 5.79e4·7-s − 9.85e4·8-s − 1.63e5·9-s + 1.25e5·10-s − 6.18e5·11-s − 3.60e5·12-s + 3.41e6·13-s − 1.15e6·14-s + 1.37e6·15-s + 6.29e5·16-s + 1.31e6·17-s + 3.26e6·18-s + 5.32e6·19-s − 1.02e7·20-s − 1.27e7·21-s + 1.23e7·22-s + 5.89e7·23-s + 2.16e7·24-s + 2.92e7·25-s − 6.82e7·26-s + 4.35e7·27-s + 9.49e7·28-s + ⋯
L(s)  = 1  − 0.441·2-s − 0.522·3-s + 0.800·4-s − 0.894·5-s + 0.231·6-s + 1.30·7-s − 1.06·8-s − 0.922·9-s + 0.395·10-s − 1.15·11-s − 0.418·12-s + 2.55·13-s − 0.575·14-s + 0.467·15-s + 0.150·16-s + 0.225·17-s + 0.407·18-s + 0.493·19-s − 0.716·20-s − 0.680·21-s + 0.511·22-s + 1.90·23-s + 0.555·24-s + 3/5·25-s − 1.12·26-s + 0.584·27-s + 1.04·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(25\)    =    \(5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(11\)
character  :  induced by $\chi_{5} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 25,\ (\ :11/2, 11/2),\ 1)$
$L(6)$  $\approx$  $1.21730$
$L(\frac12)$  $\approx$  $1.21730$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 5$,\(F_p(T)\) is a polynomial of degree 4. If $p = 5$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$ \( ( 1 + p^{5} T )^{2} \)
good2$D_{4}$ \( 1 + 5 p^{2} T - 155 p^{3} T^{2} + 5 p^{13} T^{3} + p^{22} T^{4} \)
3$D_{4}$ \( 1 + 220 T + 23530 p^{2} T^{2} + 220 p^{11} T^{3} + p^{22} T^{4} \)
7$D_{4}$ \( 1 - 57900 T + 660624350 p T^{2} - 57900 p^{11} T^{3} + p^{22} T^{4} \)
11$D_{4}$ \( 1 + 618176 T + 245194892966 T^{2} + 618176 p^{11} T^{3} + p^{22} T^{4} \)
13$D_{4}$ \( 1 - 3414260 T + 6398662197390 T^{2} - 3414260 p^{11} T^{3} + p^{22} T^{4} \)
17$D_{4}$ \( 1 - 1317940 T + 59308395866630 T^{2} - 1317940 p^{11} T^{3} + p^{22} T^{4} \)
19$D_{4}$ \( 1 - 280280 p T + 194538827137638 T^{2} - 280280 p^{12} T^{3} + p^{22} T^{4} \)
23$D_{4}$ \( 1 - 2562780 p T + 2773540471931410 T^{2} - 2562780 p^{12} T^{3} + p^{22} T^{4} \)
29$D_{4}$ \( 1 - 3246220 p T + 23426350431097358 T^{2} - 3246220 p^{12} T^{3} + p^{22} T^{4} \)
31$D_{4}$ \( 1 - 244543464 T + 34393316207729486 T^{2} - 244543464 p^{11} T^{3} + p^{22} T^{4} \)
37$D_{4}$ \( 1 - 21003220 T + 137126715218410590 T^{2} - 21003220 p^{11} T^{3} + p^{22} T^{4} \)
41$D_{4}$ \( 1 + 745743316 T + 929792912462405846 T^{2} + 745743316 p^{11} T^{3} + p^{22} T^{4} \)
43$D_{4}$ \( 1 - 629950100 T + 1840945003918927050 T^{2} - 629950100 p^{11} T^{3} + p^{22} T^{4} \)
47$D_{4}$ \( 1 + 1402061540 T + 5181805952108806370 T^{2} + 1402061540 p^{11} T^{3} + p^{22} T^{4} \)
53$D_{4}$ \( 1 - 1138320580 T - 2203723231625575330 T^{2} - 1138320580 p^{11} T^{3} + p^{22} T^{4} \)
59$D_{4}$ \( 1 - 7317515560 T + 55027608950440780118 T^{2} - 7317515560 p^{11} T^{3} + p^{22} T^{4} \)
61$D_{4}$ \( 1 + 1516425676 T + 85869525433683691566 T^{2} + 1516425676 p^{11} T^{3} + p^{22} T^{4} \)
67$D_{4}$ \( 1 - 15734290140 T + \)\(30\!\cdots\!30\)\( T^{2} - 15734290140 p^{11} T^{3} + p^{22} T^{4} \)
71$D_{4}$ \( 1 - 32938471544 T + \)\(73\!\cdots\!26\)\( T^{2} - 32938471544 p^{11} T^{3} + p^{22} T^{4} \)
73$D_{4}$ \( 1 + 29982848860 T + \)\(78\!\cdots\!10\)\( T^{2} + 29982848860 p^{11} T^{3} + p^{22} T^{4} \)
79$D_{4}$ \( 1 + 3302823120 T + \)\(12\!\cdots\!58\)\( T^{2} + 3302823120 p^{11} T^{3} + p^{22} T^{4} \)
83$D_{4}$ \( 1 - 13299102420 T + \)\(18\!\cdots\!30\)\( T^{2} - 13299102420 p^{11} T^{3} + p^{22} T^{4} \)
89$D_{4}$ \( 1 + 12674770860 T + \)\(43\!\cdots\!78\)\( T^{2} + 12674770860 p^{11} T^{3} + p^{22} T^{4} \)
97$D_{4}$ \( 1 + 3080703740 T + \)\(18\!\cdots\!70\)\( T^{2} + 3080703740 p^{11} T^{3} + p^{22} T^{4} \)
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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

−21.18229820374676738356905064732, −20.86775464936115112518076261467, −20.48001615834835926319206611091, −19.25371857873387388844211070691, −18.33336979256135259792305720362, −17.93453666412016046820795198294, −16.95574769156702062941203620297, −15.70964955423081921401802810039, −15.69645877528885658226412516390, −14.45058408175387493894649660214, −13.21307843486803838558402299942, −11.68400555027305447688553193978, −11.33877513350072406211360404469, −10.75608707608599422681900499338, −8.599382500358420294892083163862, −8.145927575509375195319117425658, −6.52847333889158114408228719657, −5.20084135453372950822890389112, −3.07033621261827739461875958973, −0.914486374519612652319967091132, 0.914486374519612652319967091132, 3.07033621261827739461875958973, 5.20084135453372950822890389112, 6.52847333889158114408228719657, 8.145927575509375195319117425658, 8.599382500358420294892083163862, 10.75608707608599422681900499338, 11.33877513350072406211360404469, 11.68400555027305447688553193978, 13.21307843486803838558402299942, 14.45058408175387493894649660214, 15.69645877528885658226412516390, 15.70964955423081921401802810039, 16.95574769156702062941203620297, 17.93453666412016046820795198294, 18.33336979256135259792305720362, 19.25371857873387388844211070691, 20.48001615834835926319206611091, 20.86775464936115112518076261467, 21.18229820374676738356905064732

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