# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.25e9·2-s + 2.00e14·3-s + 1.00e18·4-s − 4.05e20·5-s + 2.52e23·6-s + 2.66e24·7-s + 5.43e26·8-s + 2.62e28·9-s − 5.09e29·10-s + 7.79e30·11-s + 2.02e32·12-s + 1.35e32·13-s + 3.35e33·14-s − 8.14e34·15-s + 1.02e35·16-s − 2.91e36·17-s + 3.30e37·18-s − 6.13e36·19-s − 4.08e38·20-s + 5.35e38·21-s + 9.81e39·22-s − 2.47e39·23-s + 1.09e41·24-s − 9.34e39·25-s + 1.71e41·26-s + 2.43e42·27-s + 2.68e42·28-s + ⋯
 L(s)  = 1 + 1.65·2-s + 1.69·3-s + 1.74·4-s − 0.972·5-s + 2.80·6-s + 0.312·7-s + 1.24·8-s + 1.85·9-s − 1.61·10-s + 1.48·11-s + 2.95·12-s + 0.187·13-s + 0.518·14-s − 1.64·15-s + 0.309·16-s − 1.46·17-s + 3.08·18-s − 0.115·19-s − 1.70·20-s + 0.528·21-s + 2.45·22-s − 0.167·23-s + 2.09·24-s − 0.0538·25-s + 0.310·26-s + 1.45·27-s + 0.546·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(60-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+59/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1$$ Sign: $1$ Motivic weight: $$59$$ Character: $\chi_{1} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1,\ (\ :59/2),\ 1)$$

## Particular Values

 $$L(30)$$ $$\approx$$ $$7.56809$$ $$L(\frac12)$$ $$\approx$$ $$7.56809$$ $$L(\frac{61}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
good2 $$1 - 1.25e9T + 5.76e17T^{2}$$
3 $$1 - 2.00e14T + 1.41e28T^{2}$$
5 $$1 + 4.05e20T + 1.73e41T^{2}$$
7 $$1 - 2.66e24T + 7.25e49T^{2}$$
11 $$1 - 7.79e30T + 2.76e61T^{2}$$
13 $$1 - 1.35e32T + 5.28e65T^{2}$$
17 $$1 + 2.91e36T + 3.94e72T^{2}$$
19 $$1 + 6.13e36T + 2.79e75T^{2}$$
23 $$1 + 2.47e39T + 2.19e80T^{2}$$
29 $$1 + 4.74e42T + 1.91e86T^{2}$$
31 $$1 - 1.04e43T + 9.78e87T^{2}$$
37 $$1 + 7.35e45T + 3.34e92T^{2}$$
41 $$1 + 6.31e47T + 1.42e95T^{2}$$
43 $$1 + 5.26e47T + 2.36e96T^{2}$$
47 $$1 - 3.33e49T + 4.50e98T^{2}$$
53 $$1 - 7.06e50T + 5.39e101T^{2}$$
59 $$1 - 2.46e52T + 3.02e104T^{2}$$
61 $$1 - 5.32e52T + 2.16e105T^{2}$$
67 $$1 + 2.99e53T + 5.47e107T^{2}$$
71 $$1 + 4.94e54T + 1.67e109T^{2}$$
73 $$1 - 7.41e54T + 8.63e109T^{2}$$
79 $$1 + 6.76e55T + 9.12e111T^{2}$$
83 $$1 - 4.29e56T + 1.68e113T^{2}$$
89 $$1 + 4.87e56T + 1.03e115T^{2}$$
97 $$1 + 9.59e57T + 1.65e117T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$