# Properties

 Degree 4 Conductor $2^{2} \cdot 3^{2} \cdot 59^{2}$ Sign $1$ Motivic weight 3 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·2-s + 6·3-s + 12·4-s + 26·5-s − 24·6-s − 32·8-s + 27·9-s − 104·10-s + 52·11-s + 72·12-s + 66·13-s + 156·15-s + 80·16-s − 64·17-s − 108·18-s + 68·19-s + 312·20-s − 208·22-s + 116·23-s − 192·24-s + 308·25-s − 264·26-s + 108·27-s + 350·29-s − 624·30-s − 242·31-s − 192·32-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1.15·3-s + 3/2·4-s + 2.32·5-s − 1.63·6-s − 1.41·8-s + 9-s − 3.28·10-s + 1.42·11-s + 1.73·12-s + 1.40·13-s + 2.68·15-s + 5/4·16-s − 0.913·17-s − 1.41·18-s + 0.821·19-s + 3.48·20-s − 2.01·22-s + 1.05·23-s − 1.63·24-s + 2.46·25-s − 1.99·26-s + 0.769·27-s + 2.24·29-s − 3.79·30-s − 1.40·31-s − 1.06·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 125316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 125316 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$125316$$    =    $$2^{2} \cdot 3^{2} \cdot 59^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{354} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 125316,\ (\ :3/2, 3/2),\ 1)$ $L(2)$ $\approx$ $5.055880693$ $L(\frac12)$ $\approx$ $5.055880693$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;59\}$,$$F_p(T)$$ is a polynomial of degree 4. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ $$( 1 + p T )^{2}$$
3$C_1$ $$( 1 - p T )^{2}$$
59$C_1$ $$( 1 - p T )^{2}$$
good5$D_{4}$ $$1 - 26 T + 368 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4}$$
7$C_2^2$ $$1 - 130 T^{2} + p^{6} T^{4}$$
11$D_{4}$ $$1 - 52 T + 2522 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 - 66 T + 2984 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 + 64 T + 9014 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4}$$
19$C_2$ $$( 1 - 34 T + p^{3} T^{2} )^{2}$$
23$D_{4}$ $$1 - 116 T + 25862 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 - 350 T + 75272 T^{2} - 350 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 + 242 T + 59484 T^{2} + 242 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 + 638 T + 200568 T^{2} + 638 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 + 616 T + 216182 T^{2} + 616 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 - 768 T + 303206 T^{2} - 768 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 - 44 T + 134486 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 - 90 T + 295648 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 - 1250 T + 817608 T^{2} - 1250 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 - 1460 T + 1124430 T^{2} - 1460 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 - 162 T - 100196 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 - 284 T + 203334 T^{2} - 284 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 + 512 T + 108318 T^{2} + 512 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 - 376 T + 708902 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 + 492 T + 1057354 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 + 1088 T + 1259382 T^{2} + 1088 p^{3} T^{3} + p^{6} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}