# Properties

 Degree 2 Conductor $1$ Sign $1$ Motivic weight 67 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.26e10·2-s − 1.28e15·3-s + 3.67e20·4-s − 1.61e23·5-s − 2.91e25·6-s + 2.93e28·7-s + 4.99e30·8-s − 9.10e31·9-s − 3.65e33·10-s + 1.03e35·11-s − 4.71e35·12-s − 9.30e35·13-s + 6.66e38·14-s + 2.07e38·15-s + 5.90e40·16-s + 2.52e41·17-s − 2.06e42·18-s − 4.10e41·19-s − 5.92e43·20-s − 3.77e43·21-s + 2.36e45·22-s − 7.28e44·23-s − 6.40e45·24-s − 4.17e46·25-s − 2.11e46·26-s + 2.36e47·27-s + 1.07e49·28-s + ⋯
 L(s)  = 1 + 1.86·2-s − 0.133·3-s + 2.49·4-s − 0.619·5-s − 0.249·6-s + 1.43·7-s + 2.78·8-s − 0.982·9-s − 1.15·10-s + 1.34·11-s − 0.332·12-s − 0.0448·13-s + 2.68·14-s + 0.0825·15-s + 2.71·16-s + 1.51·17-s − 1.83·18-s − 0.0596·19-s − 1.54·20-s − 0.191·21-s + 2.52·22-s − 0.175·23-s − 0.371·24-s − 0.616·25-s − 0.0837·26-s + 0.264·27-s + 3.57·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(68-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+67/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1$$ $$\varepsilon$$ = $1$ motivic weight = $$67$$ character : $\chi_{1} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 1,\ (\ :67/2),\ 1)$$ $$L(34)$$ $$\approx$$ $$6.502194688$$ $$L(\frac12)$$ $$\approx$$ $$6.502194688$$ $$L(\frac{69}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where,$$F_p(T)$$ is a polynomial of degree 2.
$p$$F_p(T)$
good2 $$1 - 2.26e10T + 1.47e20T^{2}$$
3 $$1 + 1.28e15T + 9.27e31T^{2}$$
5 $$1 + 1.61e23T + 6.77e46T^{2}$$
7 $$1 - 2.93e28T + 4.18e56T^{2}$$
11 $$1 - 1.03e35T + 5.93e69T^{2}$$
13 $$1 + 9.30e35T + 4.30e74T^{2}$$
17 $$1 - 2.52e41T + 2.75e82T^{2}$$
19 $$1 + 4.10e41T + 4.74e85T^{2}$$
23 $$1 + 7.28e44T + 1.72e91T^{2}$$
29 $$1 + 8.66e48T + 9.56e97T^{2}$$
31 $$1 - 5.42e49T + 8.34e99T^{2}$$
37 $$1 - 2.44e52T + 1.17e105T^{2}$$
41 $$1 - 8.84e52T + 1.13e108T^{2}$$
43 $$1 + 3.35e54T + 2.76e109T^{2}$$
47 $$1 + 1.34e56T + 1.07e112T^{2}$$
53 $$1 + 3.87e57T + 3.36e115T^{2}$$
59 $$1 + 2.62e59T + 4.43e118T^{2}$$
61 $$1 + 4.77e59T + 4.14e119T^{2}$$
67 $$1 + 2.36e61T + 2.22e122T^{2}$$
71 $$1 - 2.36e61T + 1.08e124T^{2}$$
73 $$1 - 7.86e61T + 6.96e124T^{2}$$
79 $$1 + 2.74e63T + 1.38e127T^{2}$$
83 $$1 - 9.16e63T + 3.78e128T^{2}$$
89 $$1 + 3.36e65T + 4.06e130T^{2}$$
97 $$1 - 6.50e66T + 1.29e133T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}