# Properties

 Degree 2 Conductor $1$ Sign $-1$ Motivic weight 69 Primitive yes Self-dual yes Analytic rank 1

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## Dirichlet series

 L(s)  = 1 + 2.12e10·2-s + 3.67e16·3-s − 1.38e20·4-s − 2.01e24·5-s + 7.80e26·6-s + 2.05e29·7-s − 1.54e31·8-s + 5.14e32·9-s − 4.28e34·10-s − 1.09e36·11-s − 5.08e36·12-s − 2.46e37·13-s + 4.37e39·14-s − 7.40e40·15-s − 2.47e41·16-s − 2.55e42·17-s + 1.09e43·18-s − 1.09e44·19-s + 2.79e44·20-s + 7.55e45·21-s − 2.31e46·22-s − 8.34e45·23-s − 5.68e47·24-s + 2.37e48·25-s − 5.24e47·26-s − 1.17e49·27-s − 2.85e49·28-s + ⋯
 L(s)  = 1 + 0.874·2-s + 1.27·3-s − 0.234·4-s − 1.54·5-s + 1.11·6-s + 1.43·7-s − 1.08·8-s + 0.616·9-s − 1.35·10-s − 1.28·11-s − 0.298·12-s − 0.0914·13-s + 1.25·14-s − 1.96·15-s − 0.710·16-s − 0.903·17-s + 0.539·18-s − 0.837·19-s + 0.363·20-s + 1.82·21-s − 1.12·22-s − 0.0874·23-s − 1.37·24-s + 1.39·25-s − 0.0799·26-s − 0.487·27-s − 0.337·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\,\Lambda(70-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+69/2) \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1$$ $$\varepsilon$$ = $-1$ motivic weight = $$69$$ character : $\chi_{1} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 1,\ (\ :69/2),\ -1)$ $L(35)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{71}{2})$ 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 - 2.12e10T + 5.90e20T^{2}$$
3 $$1 - 3.67e16T + 8.34e32T^{2}$$
5 $$1 + 2.01e24T + 1.69e48T^{2}$$
7 $$1 - 2.05e29T + 2.05e58T^{2}$$
11 $$1 + 1.09e36T + 7.17e71T^{2}$$
13 $$1 + 2.46e37T + 7.27e76T^{2}$$
17 $$1 + 2.55e42T + 7.96e84T^{2}$$
19 $$1 + 1.09e44T + 1.71e88T^{2}$$
23 $$1 + 8.34e45T + 9.10e93T^{2}$$
29 $$1 + 4.43e49T + 8.04e100T^{2}$$
31 $$1 + 2.07e51T + 8.01e102T^{2}$$
37 $$1 - 2.01e54T + 1.60e108T^{2}$$
41 $$1 + 1.70e55T + 1.91e111T^{2}$$
43 $$1 + 1.46e55T + 5.12e112T^{2}$$
47 $$1 + 7.82e57T + 2.37e115T^{2}$$
53 $$1 - 4.83e59T + 9.44e118T^{2}$$
59 $$1 + 1.34e61T + 1.54e122T^{2}$$
61 $$1 - 3.06e61T + 1.54e123T^{2}$$
67 $$1 - 5.79e62T + 9.98e125T^{2}$$
71 $$1 - 5.96e63T + 5.45e127T^{2}$$
73 $$1 + 2.58e63T + 3.70e128T^{2}$$
79 $$1 + 1.36e65T + 8.63e130T^{2}$$
83 $$1 - 4.54e65T + 2.60e132T^{2}$$
89 $$1 + 6.95e66T + 3.22e134T^{2}$$
97 $$1 - 3.36e68T + 1.22e137T^{2}$$
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\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−15.33142775789665464152808433329, −14.62144898066989195572976184089, −13.08286449419592919848122442194, −11.34282676636948805795166073956, −8.570633730192148129111991215485, −7.81097344317321505890215721680, −4.84086002681136528295761321043, −3.84643910427900934906866316490, −2.44941283318093265885292563093, 0, 2.44941283318093265885292563093, 3.84643910427900934906866316490, 4.84086002681136528295761321043, 7.81097344317321505890215721680, 8.570633730192148129111991215485, 11.34282676636948805795166073956, 13.08286449419592919848122442194, 14.62144898066989195572976184089, 15.33142775789665464152808433329