# Properties

 Degree 4 Conductor $13^{2}$ Sign $1$ Motivic weight 3 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·2-s − 2·3-s − 2·4-s − 6·6-s − 24·7-s − 15·8-s + 27·9-s + 24·11-s + 4·12-s + 91·13-s − 72·14-s − 15·16-s − 117·17-s + 81·18-s − 198·19-s + 48·21-s + 72·22-s + 78·23-s + 30·24-s + 247·25-s + 273·26-s − 154·27-s + 48·28-s + 141·29-s − 120·32-s − 48·33-s − 351·34-s + ⋯
 L(s)  = 1 + 1.06·2-s − 0.384·3-s − 1/4·4-s − 0.408·6-s − 1.29·7-s − 0.662·8-s + 9-s + 0.657·11-s + 0.0962·12-s + 1.94·13-s − 1.37·14-s − 0.234·16-s − 1.66·17-s + 1.06·18-s − 2.39·19-s + 0.498·21-s + 0.697·22-s + 0.707·23-s + 0.255·24-s + 1.97·25-s + 2.05·26-s − 1.09·27-s + 0.323·28-s + 0.902·29-s − 0.662·32-s − 0.253·33-s − 1.77·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$169$$    =    $$13^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{13} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 169,\ (\ :3/2, 3/2),\ 1)$$ $$L(2)$$ $$\approx$$ $$1.10692$$ $$L(\frac12)$$ $$\approx$$ $$1.10692$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 13$,$$F_p(T)$$ is a polynomial of degree 4. If $p = 13$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad13$C_2$ $$1 - 7 p T + p^{3} T^{2}$$
good2$C_2^2$ $$1 - 3 T + 11 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4}$$
3$C_2^2$ $$1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4}$$
5$C_2^2$ $$1 - 247 T^{2} + p^{6} T^{4}$$
7$C_2^2$ $$1 + 24 T + 535 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4}$$
11$C_2^2$ $$1 - 24 T + 1523 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4}$$
17$C_2^2$ $$1 + 117 T + 8776 T^{2} + 117 p^{3} T^{3} + p^{6} T^{4}$$
19$C_2^2$ $$1 + 198 T + 19927 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4}$$
23$C_2^2$ $$1 - 78 T - 6083 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4}$$
29$C_2^2$ $$1 - 141 T - 4508 T^{2} - 141 p^{3} T^{3} + p^{6} T^{4}$$
31$C_2$ $$( 1 - 308 T + p^{3} T^{2} )( 1 + 308 T + p^{3} T^{2} )$$
37$C_2^2$ $$1 + 249 T + 71320 T^{2} + 249 p^{3} T^{3} + p^{6} T^{4}$$
41$C_2^2$ $$1 - 471 T + 142868 T^{2} - 471 p^{3} T^{3} + p^{6} T^{4}$$
43$C_2^2$ $$1 + 104 T - 68691 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4}$$
47$C_2^2$ $$1 - 116818 T^{2} + p^{6} T^{4}$$
53$C_2$ $$( 1 - 93 T + p^{3} T^{2} )^{2}$$
59$C_2^2$ $$1 + 492 T + 286067 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4}$$
61$C_2^2$ $$1 + 145 T - 205956 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4}$$
67$C_2^2$ $$1 + 1362 T + 919111 T^{2} + 1362 p^{3} T^{3} + p^{6} T^{4}$$
71$C_2^2$ $$1 - 1830 T + 1474211 T^{2} - 1830 p^{3} T^{3} + p^{6} T^{4}$$
73$C_2^2$ $$1 - 567359 T^{2} + p^{6} T^{4}$$
79$C_2$ $$( 1 - 1276 T + p^{3} T^{2} )^{2}$$
83$C_2^2$ $$1 - 519766 T^{2} + p^{6} T^{4}$$
89$C_2^2$ $$1 + 1692 T + 1659257 T^{2} + 1692 p^{3} T^{3} + p^{6} T^{4}$$
97$C_2^2$ $$1 - 348 T + 953041 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}