# Properties

 Degree 4 Conductor $7^{2}$ Sign $1$ Motivic weight 5 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 9·2-s − 6·3-s + 11·4-s − 18·5-s − 54·6-s + 98·7-s − 243·8-s + 54·9-s − 162·10-s + 396·11-s − 66·12-s − 350·13-s + 882·14-s + 108·15-s − 1.38e3·16-s + 1.80e3·17-s + 486·18-s − 3.26e3·19-s − 198·20-s − 588·21-s + 3.56e3·22-s + 2.08e3·23-s + 1.45e3·24-s − 4.58e3·25-s − 3.15e3·26-s − 1.89e3·27-s + 1.07e3·28-s + ⋯
 L(s)  = 1 + 1.59·2-s − 0.384·3-s + 0.343·4-s − 0.321·5-s − 0.612·6-s + 0.755·7-s − 1.34·8-s + 2/9·9-s − 0.512·10-s + 0.986·11-s − 0.132·12-s − 0.574·13-s + 1.20·14-s + 0.123·15-s − 1.35·16-s + 1.51·17-s + 0.353·18-s − 2.07·19-s − 0.110·20-s − 0.290·21-s + 1.56·22-s + 0.823·23-s + 0.516·24-s − 1.46·25-s − 0.913·26-s − 0.498·27-s + 0.259·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$49$$    =    $$7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : induced by $\chi_{7} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 49,\ (\ :5/2, 5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$1.71318$$ $$L(\frac12)$$ $$\approx$$ $$1.71318$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 7$,$$F_p(T)$$ is a polynomial of degree 4. If $p = 7$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad7$C_1$ $$( 1 - p^{2} T )^{2}$$
good2$D_{4}$ $$1 - 9 T + 35 p T^{2} - 9 p^{5} T^{3} + p^{10} T^{4}$$
3$D_{4}$ $$1 + 2 p T - 2 p^{2} T^{2} + 2 p^{6} T^{3} + p^{10} T^{4}$$
5$D_{4}$ $$1 + 18 T + 4906 T^{2} + 18 p^{5} T^{3} + p^{10} T^{4}$$
11$D_{4}$ $$1 - 36 p T + 142198 T^{2} - 36 p^{6} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 + 350 T + 546978 T^{2} + 350 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 - 1800 T + 3567406 T^{2} - 1800 p^{5} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 + 3266 T + 7614270 T^{2} + 3266 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 - 2088 T + 9365230 T^{2} - 2088 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 - 6696 T + 51326470 T^{2} - 6696 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 + 20 T + 53103102 T^{2} + 20 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 - 6232 T + 144242070 T^{2} - 6232 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 + 6048 T + 223864366 T^{2} + 6048 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 + 3020 T - 30383466 T^{2} + 3020 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 - 11700 T + 292735582 T^{2} - 11700 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 - 9468 T + 858185230 T^{2} - 9468 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 + 43938 T + 1852599934 T^{2} + 43938 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 + 64754 T + 2408321418 T^{2} + 64754 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 - 24784 T + 2799959190 T^{2} - 24784 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 - 97416 T + 5729557966 T^{2} - 97416 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 - 17452 T + 3828622374 T^{2} - 17452 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 - 51256 T + 3645565854 T^{2} - 51256 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 - 117558 T + 7798161502 T^{2} - 117558 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 - 84276 T + 5915697430 T^{2} - 84276 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 - 20776 T + 16174049358 T^{2} - 20776 p^{5} T^{3} + p^{10} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}