# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 59$ Sign $1$ Motivic weight 5 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 4·2-s − 9·3-s + 16·4-s + 10·5-s − 36·6-s + 144·7-s + 64·8-s + 81·9-s + 40·10-s + 668·11-s − 144·12-s − 270·13-s + 576·14-s − 90·15-s + 256·16-s − 758·17-s + 324·18-s + 868·19-s + 160·20-s − 1.29e3·21-s + 2.67e3·22-s + 784·23-s − 576·24-s − 3.02e3·25-s − 1.08e3·26-s − 729·27-s + 2.30e3·28-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.178·5-s − 0.408·6-s + 1.11·7-s + 0.353·8-s + 1/3·9-s + 0.126·10-s + 1.66·11-s − 0.288·12-s − 0.443·13-s + 0.785·14-s − 0.103·15-s + 1/4·16-s − 0.636·17-s + 0.235·18-s + 0.551·19-s + 0.0894·20-s − 0.641·21-s + 1.17·22-s + 0.309·23-s − 0.204·24-s − 0.967·25-s − 0.313·26-s − 0.192·27-s + 0.555·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$354$$    =    $$2 \cdot 3 \cdot 59$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : $\chi_{354} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 354,\ (\ :5/2),\ 1)$ $L(3)$ $\approx$ $3.678629785$ $L(\frac12)$ $\approx$ $3.678629785$ $L(\frac{7}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;59\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - p^{2} T$$
3 $$1 + p^{2} T$$
59 $$1 + p^{2} T$$
good5 $$1 - 2 p T + p^{5} T^{2}$$
7 $$1 - 144 T + p^{5} T^{2}$$
11 $$1 - 668 T + p^{5} T^{2}$$
13 $$1 + 270 T + p^{5} T^{2}$$
17 $$1 + 758 T + p^{5} T^{2}$$
19 $$1 - 868 T + p^{5} T^{2}$$
23 $$1 - 784 T + p^{5} T^{2}$$
29 $$1 + 4574 T + p^{5} T^{2}$$
31 $$1 - 8948 T + p^{5} T^{2}$$
37 $$1 + 670 T + p^{5} T^{2}$$
41 $$1 + 7934 T + p^{5} T^{2}$$
43 $$1 - 4884 T + p^{5} T^{2}$$
47 $$1 - 24280 T + p^{5} T^{2}$$
53 $$1 - 28962 T + p^{5} T^{2}$$
61 $$1 - 30490 T + p^{5} T^{2}$$
67 $$1 + 30764 T + p^{5} T^{2}$$
71 $$1 - 22452 T + p^{5} T^{2}$$
73 $$1 + 20966 T + p^{5} T^{2}$$
79 $$1 - 70520 T + p^{5} T^{2}$$
83 $$1 - 29756 T + p^{5} T^{2}$$
89 $$1 + 16470 T + p^{5} T^{2}$$
97 $$1 - 18506 T + p^{5} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}