# Properties

 Degree 2 Conductor $1$ Sign $1$ Motivic weight 103 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 5.46e15·2-s − 1.30e24·3-s + 1.96e31·4-s − 5.30e35·5-s + 7.10e39·6-s + 2.61e43·7-s − 5.20e46·8-s − 1.22e49·9-s + 2.89e51·10-s + 4.50e53·11-s − 2.55e55·12-s + 2.00e57·13-s − 1.43e59·14-s + 6.90e59·15-s + 8.46e61·16-s + 3.55e63·17-s + 6.67e64·18-s + 1.06e66·19-s − 1.04e67·20-s − 3.40e67·21-s − 2.45e69·22-s + 1.51e70·23-s + 6.77e70·24-s − 7.04e71·25-s − 1.09e73·26-s + 3.40e73·27-s + 5.15e74·28-s + ⋯
 L(s)  = 1 − 1.71·2-s − 0.348·3-s + 1.93·4-s − 0.534·5-s + 0.597·6-s + 0.786·7-s − 1.61·8-s − 0.878·9-s + 0.916·10-s + 1.05·11-s − 0.676·12-s + 0.857·13-s − 1.34·14-s + 0.186·15-s + 0.823·16-s + 1.52·17-s + 1.50·18-s + 1.47·19-s − 1.03·20-s − 0.274·21-s − 1.80·22-s + 1.12·23-s + 0.562·24-s − 0.714·25-s − 1.47·26-s + 0.655·27-s + 1.52·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \,\Lambda(104-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+51.5) \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1$$ $$\varepsilon$$ = $1$ motivic weight = $$103$$ character : $\chi_{1} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 1,\ (\ :103/2),\ 1)$ $L(52)$ $\approx$ $0.8749951250$ $L(\frac12)$ $\approx$ $0.8749951250$ $L(\frac{105}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, $$F_p$$ is a polynomial of degree 2.
$p$$F_p$
good2 $$1 + 5.46e15T + 1.01e31T^{2}$$
3 $$1 + 1.30e24T + 1.39e49T^{2}$$
5 $$1 + 5.30e35T + 9.86e71T^{2}$$
7 $$1 - 2.61e43T + 1.10e87T^{2}$$
11 $$1 - 4.50e53T + 1.83e107T^{2}$$
13 $$1 - 2.00e57T + 5.44e114T^{2}$$
17 $$1 - 3.55e63T + 5.44e126T^{2}$$
19 $$1 - 1.06e66T + 5.14e131T^{2}$$
23 $$1 - 1.51e70T + 1.81e140T^{2}$$
29 $$1 + 1.95e75T + 4.23e150T^{2}$$
31 $$1 + 6.33e76T + 4.07e153T^{2}$$
37 $$1 + 3.73e80T + 3.34e161T^{2}$$
41 $$1 - 1.79e83T + 1.30e166T^{2}$$
43 $$1 + 4.86e83T + 1.76e168T^{2}$$
47 $$1 + 1.62e86T + 1.68e172T^{2}$$
53 $$1 - 7.08e88T + 3.98e177T^{2}$$
59 $$1 + 2.20e90T + 2.49e182T^{2}$$
61 $$1 - 4.53e91T + 7.74e183T^{2}$$
67 $$1 - 1.42e94T + 1.21e188T^{2}$$
71 $$1 - 1.63e95T + 4.78e190T^{2}$$
73 $$1 + 1.05e96T + 8.36e191T^{2}$$
79 $$1 + 7.93e97T + 2.85e195T^{2}$$
83 $$1 - 5.44e98T + 4.62e197T^{2}$$
89 $$1 + 3.14e100T + 6.12e200T^{2}$$
97 $$1 - 6.67e101T + 4.33e204T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}