# Properties

 Degree 4 Conductor $3^{4} \cdot 7^{2} \cdot 11^{4}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 2

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·2-s + 4·4-s − 2·5-s − 2·7-s − 3·8-s + 6·10-s − 2·13-s + 6·14-s + 3·16-s − 5·17-s + 8·19-s − 8·20-s + 23-s − 7·25-s + 6·26-s − 8·28-s − 7·29-s − 4·31-s − 6·32-s + 15·34-s + 4·35-s + 4·37-s − 24·38-s + 6·40-s + 5·43-s − 3·46-s + 11·47-s + ⋯
 L(s)  = 1 − 2.12·2-s + 2·4-s − 0.894·5-s − 0.755·7-s − 1.06·8-s + 1.89·10-s − 0.554·13-s + 1.60·14-s + 3/4·16-s − 1.21·17-s + 1.83·19-s − 1.78·20-s + 0.208·23-s − 7/5·25-s + 1.17·26-s − 1.51·28-s − 1.29·29-s − 0.718·31-s − 1.06·32-s + 2.57·34-s + 0.676·35-s + 0.657·37-s − 3.89·38-s + 0.948·40-s + 0.762·43-s − 0.442·46-s + 1.60·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$58110129$$    =    $$3^{4} \cdot 7^{2} \cdot 11^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{7623} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 2 Selberg data = $(4,\ 58110129,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;7,\;11\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3 $$1$$
7$C_1$ $$( 1 + T )^{2}$$
11 $$1$$
good2$C_2^2$ $$1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
5$C_2$ $$( 1 + T + p T^{2} )^{2}$$
13$D_{4}$ $$1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 5 T + 9 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 - 8 T + 49 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 7 T + 69 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 + 4 T + 61 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
41$C_2^2$ $$1 - 43 T^{2} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 5 T - 9 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 - 11 T + 123 T^{2} - 11 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 7 T + 117 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 11 T + 117 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 - 13 T + 163 T^{2} - 13 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + T + 73 T^{2} + p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 - 21 T + 221 T^{2} - 21 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 24 T + 285 T^{2} - 24 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 15 T + 213 T^{2} + 15 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 8 T + 137 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 7 T + 179 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 7 T + p T^{2} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}