# Properties

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

# Origins of factors

## 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$D_{4}$ $$1 + 3080703740 T +$$$$18\!\cdots\!70$$$$T^{2} + 3080703740 p^{11} T^{3} + p^{22} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}