# Properties

 Degree 2 Conductor 619 Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.39·2-s + 2.10·3-s − 0.0457·4-s + 3.90·5-s + 2.94·6-s − 1.82·7-s − 2.85·8-s + 1.43·9-s + 5.46·10-s + 4.06·11-s − 0.0963·12-s − 2.56·13-s − 2.54·14-s + 8.22·15-s − 3.90·16-s + 2.96·17-s + 2.00·18-s − 3.33·19-s − 0.178·20-s − 3.83·21-s + 5.68·22-s − 9.09·23-s − 6.02·24-s + 10.2·25-s − 3.58·26-s − 3.30·27-s + 0.0834·28-s + ⋯
 L(s)  = 1 + 0.988·2-s + 1.21·3-s − 0.0228·4-s + 1.74·5-s + 1.20·6-s − 0.688·7-s − 1.01·8-s + 0.477·9-s + 1.72·10-s + 1.22·11-s − 0.0278·12-s − 0.712·13-s − 0.680·14-s + 2.12·15-s − 0.976·16-s + 0.717·17-s + 0.471·18-s − 0.764·19-s − 0.0400·20-s − 0.837·21-s + 1.21·22-s − 1.89·23-s − 1.22·24-s + 2.05·25-s − 0.703·26-s − 0.635·27-s + 0.0157·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$619$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{619} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 619,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $3.67966$ $L(\frac12)$ $\approx$ $3.67966$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 619$, $F_p(T) = 1 - a_p T + p T^2 .$If $p = 619$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad619 $$1 - T$$
good2 $$1 - 1.39T + 2T^{2}$$
3 $$1 - 2.10T + 3T^{2}$$
5 $$1 - 3.90T + 5T^{2}$$
7 $$1 + 1.82T + 7T^{2}$$
11 $$1 - 4.06T + 11T^{2}$$
13 $$1 + 2.56T + 13T^{2}$$
17 $$1 - 2.96T + 17T^{2}$$
19 $$1 + 3.33T + 19T^{2}$$
23 $$1 + 9.09T + 23T^{2}$$
29 $$1 - 7.13T + 29T^{2}$$
31 $$1 - 3.95T + 31T^{2}$$
37 $$1 + 2.67T + 37T^{2}$$
41 $$1 + 9.73T + 41T^{2}$$
43 $$1 - 3.74T + 43T^{2}$$
47 $$1 + 6.42T + 47T^{2}$$
53 $$1 - 1.59T + 53T^{2}$$
59 $$1 + 11.4T + 59T^{2}$$
61 $$1 - 11.9T + 61T^{2}$$
67 $$1 + 1.47T + 67T^{2}$$
71 $$1 - 15.1T + 71T^{2}$$
73 $$1 + 7.51T + 73T^{2}$$
79 $$1 - 1.32T + 79T^{2}$$
83 $$1 - 9.23T + 83T^{2}$$
89 $$1 - 6.11T + 89T^{2}$$
97 $$1 - 14.0T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}