# Properties

 Degree 2 Conductor 5077 Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.65·2-s − 0.258·3-s + 5.04·4-s − 1.45·5-s + 0.686·6-s − 0.380·7-s − 8.06·8-s − 2.93·9-s + 3.84·10-s + 3.71·11-s − 1.30·12-s − 1.35·13-s + 1.01·14-s + 0.375·15-s + 11.3·16-s + 6.71·17-s + 7.78·18-s + 6.09·19-s − 7.31·20-s + 0.0984·21-s − 9.84·22-s + 7.60·23-s + 2.08·24-s − 2.89·25-s + 3.58·26-s + 1.53·27-s − 1.91·28-s + ⋯
 L(s)  = 1 − 1.87·2-s − 0.149·3-s + 2.52·4-s − 0.648·5-s + 0.280·6-s − 0.143·7-s − 2.85·8-s − 0.977·9-s + 1.21·10-s + 1.11·11-s − 0.376·12-s − 0.374·13-s + 0.269·14-s + 0.0968·15-s + 2.83·16-s + 1.62·17-s + 1.83·18-s + 1.39·19-s − 1.63·20-s + 0.0214·21-s − 2.09·22-s + 1.58·23-s + 0.426·24-s − 0.579·25-s + 0.702·26-s + 0.295·27-s − 0.362·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$5077$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{5077} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 5077,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $0.7449863547$ $L(\frac12)$ $\approx$ $0.7449863547$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5077$, $F_p(T) = 1 - a_p T + p T^2 .$If $p = 5077$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5077 $$1+O(T)$$
good2 $$1 + 2.65T + 2T^{2}$$
3 $$1 + 0.258T + 3T^{2}$$
5 $$1 + 1.45T + 5T^{2}$$
7 $$1 + 0.380T + 7T^{2}$$
11 $$1 - 3.71T + 11T^{2}$$
13 $$1 + 1.35T + 13T^{2}$$
17 $$1 - 6.71T + 17T^{2}$$
19 $$1 - 6.09T + 19T^{2}$$
23 $$1 - 7.60T + 23T^{2}$$
29 $$1 - 2.33T + 29T^{2}$$
31 $$1 - 10.0T + 31T^{2}$$
37 $$1 - 9.80T + 37T^{2}$$
41 $$1 - 4.45T + 41T^{2}$$
43 $$1 - 1.19T + 43T^{2}$$
47 $$1 - 3.67T + 47T^{2}$$
53 $$1 - 7.76T + 53T^{2}$$
59 $$1 + 9.70T + 59T^{2}$$
61 $$1 + 5.93T + 61T^{2}$$
67 $$1 + 6.73T + 67T^{2}$$
71 $$1 + 12.6T + 71T^{2}$$
73 $$1 + 11.2T + 73T^{2}$$
79 $$1 + 13.4T + 79T^{2}$$
83 $$1 + 4.18T + 83T^{2}$$
89 $$1 - 1.66T + 89T^{2}$$
97 $$1 - 8.53T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}