# Properties

 Degree 4 Conductor $7^{2}$ Sign $1$ Motivic weight 3 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s − 7·3-s + 8·4-s − 7·5-s + 14·6-s + 28·7-s − 40·8-s + 27·9-s + 14·10-s + 5·11-s − 56·12-s − 28·13-s − 56·14-s + 49·15-s + 80·16-s + 21·17-s − 54·18-s − 49·19-s − 56·20-s − 196·21-s − 10·22-s + 159·23-s + 280·24-s + 125·25-s + 56·26-s − 224·27-s + 224·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.34·3-s + 4-s − 0.626·5-s + 0.952·6-s + 1.51·7-s − 1.76·8-s + 9-s + 0.442·10-s + 0.137·11-s − 1.34·12-s − 0.597·13-s − 1.06·14-s + 0.843·15-s + 5/4·16-s + 0.299·17-s − 0.707·18-s − 0.591·19-s − 0.626·20-s − 2.03·21-s − 0.0969·22-s + 1.44·23-s + 2.38·24-s + 25-s + 0.422·26-s − 1.59·27-s + 1.51·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$49$$    =    $$7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{7} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 49,\ (\ :3/2, 3/2),\ 1)$ $L(2)$ $\approx$ $0.357271$ $L(\frac12)$ $\approx$ $0.357271$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 7$, $$F_p$$ is a polynomial of degree 4. If $p = 7$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad7$C_2$ $$1 - 4 p T + p^{3} T^{2}$$
good2$C_2^2$ $$1 + p T - p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4}$$
3$C_2^2$ $$1 + 7 T + 22 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4}$$
5$C_2^2$ $$1 + 7 T - 76 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4}$$
11$C_2^2$ $$1 - 5 T - 1306 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4}$$
13$C_2$ $$( 1 + 14 T + p^{3} T^{2} )^{2}$$
17$C_2^2$ $$1 - 21 T - 4472 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4}$$
19$C_2^2$ $$1 + 49 T - 4458 T^{2} + 49 p^{3} T^{3} + p^{6} T^{4}$$
23$C_2^2$ $$1 - 159 T + 13114 T^{2} - 159 p^{3} T^{3} + p^{6} T^{4}$$
29$C_2$ $$( 1 - 2 p T + p^{3} T^{2} )^{2}$$
31$C_2^2$ $$1 + 147 T - 8182 T^{2} + 147 p^{3} T^{3} + p^{6} T^{4}$$
37$C_2^2$ $$1 + 219 T - 2692 T^{2} + 219 p^{3} T^{3} + p^{6} T^{4}$$
41$C_2$ $$( 1 - 350 T + p^{3} T^{2} )^{2}$$
43$C_2$ $$( 1 + 124 T + p^{3} T^{2} )^{2}$$
47$C_2^2$ $$1 + 525 T + 171802 T^{2} + 525 p^{3} T^{3} + p^{6} T^{4}$$
53$C_2^2$ $$1 + 303 T - 57068 T^{2} + 303 p^{3} T^{3} + p^{6} T^{4}$$
59$C_2^2$ $$1 - 105 T - 194354 T^{2} - 105 p^{3} T^{3} + p^{6} T^{4}$$
61$C_2^2$ $$1 - 413 T - 56412 T^{2} - 413 p^{3} T^{3} + p^{6} T^{4}$$
67$C_2^2$ $$1 + 415 T - 128538 T^{2} + 415 p^{3} T^{3} + p^{6} T^{4}$$
71$C_2$ $$( 1 + 432 T + p^{3} T^{2} )^{2}$$
73$C_2^2$ $$1 - 1113 T + 849752 T^{2} - 1113 p^{3} T^{3} + p^{6} T^{4}$$
79$C_2^2$ $$1 - 103 T - 482430 T^{2} - 103 p^{3} T^{3} + p^{6} T^{4}$$
83$C_2$ $$( 1 - 1092 T + p^{3} T^{2} )^{2}$$
89$C_2^2$ $$1 - 329 T - 596728 T^{2} - 329 p^{3} T^{3} + p^{6} T^{4}$$
97$C_2$ $$( 1 + 882 T + p^{3} T^{2} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}