# Properties

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

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·2-s − 2·3-s + 4·4-s + 6·6-s − 3·8-s + 3·9-s − 8·12-s − 5·13-s + 3·16-s + 3·17-s − 9·18-s − 6·19-s + 6·23-s + 6·24-s + 7·25-s + 15·26-s − 10·27-s − 3·29-s − 6·32-s − 9·34-s + 12·36-s + 15·37-s + 18·38-s + 10·39-s − 9·41-s − 8·43-s − 18·46-s + ⋯
 L(s)  = 1 − 2.12·2-s − 1.15·3-s + 2·4-s + 2.44·6-s − 1.06·8-s + 9-s − 2.30·12-s − 1.38·13-s + 3/4·16-s + 0.727·17-s − 2.12·18-s − 1.37·19-s + 1.25·23-s + 1.22·24-s + 7/5·25-s + 2.94·26-s − 1.92·27-s − 0.557·29-s − 1.06·32-s − 1.54·34-s + 2·36-s + 2.46·37-s + 2.91·38-s + 1.60·39-s − 1.40·41-s − 1.21·43-s − 2.65·46-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$169$$    =    $$13^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{169} (1, \cdot )$ Sato-Tate : $E_6$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 169,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.09049039083$ $L(\frac12)$ $\approx$ $0.09049039083$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 13$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 13$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad13$C_2$ $$1 + 5 T + p T^{2}$$
good2$V_4$ $$1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
3$V_4$ $$1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4}$$
5$V_4$ $$1 - 7 T^{2} + p^{2} T^{4}$$
7$V_4$ $$1 + p T^{2} + p^{2} T^{4}$$
11$V_4$ $$1 + p T^{2} + p^{2} T^{4}$$
17$V_4$ $$1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
19$C_2$ $$( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} )$$
23$V_4$ $$1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
29$V_4$ $$1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
31$V_4$ $$1 - 50 T^{2} + p^{2} T^{4}$$
37$V_4$ $$1 - 15 T + 112 T^{2} - 15 p T^{3} + p^{2} T^{4}$$
41$V_4$ $$1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
43$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} )$$
47$V_4$ $$1 - 82 T^{2} + p^{2} T^{4}$$
53$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
59$V_4$ $$1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
61$C_2$ $$( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
67$C_2$ $$( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
71$V_4$ $$1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
73$C_2$ $$( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} )$$
79$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
83$V_4$ $$1 + 26 T^{2} + p^{2} T^{4}$$
89$V_4$ $$1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
97$V_4$ $$1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}