# Properties

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

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## Dirichlet series

 L(s)  = 1 − 6.55e8·2-s + 1.58e14·3-s − 1.46e17·4-s + 7.23e20·5-s − 1.04e23·6-s + 1.08e25·7-s + 4.74e26·8-s + 1.10e28·9-s − 4.74e29·10-s − 2.58e30·11-s − 2.32e31·12-s − 5.56e32·13-s − 7.12e33·14-s + 1.14e35·15-s − 2.26e35·16-s + 1.35e35·17-s − 7.26e36·18-s + 4.79e37·19-s − 1.06e38·20-s + 1.72e39·21-s + 1.69e39·22-s − 3.11e39·23-s + 7.52e40·24-s + 3.50e41·25-s + 3.65e41·26-s − 4.84e41·27-s − 1.59e42·28-s + ⋯
 L(s)  = 1 − 0.863·2-s + 1.33·3-s − 0.254·4-s + 1.73·5-s − 1.15·6-s + 1.27·7-s + 1.08·8-s + 0.784·9-s − 1.50·10-s − 0.491·11-s − 0.339·12-s − 0.766·13-s − 1.10·14-s + 2.32·15-s − 0.681·16-s + 0.0681·17-s − 0.677·18-s + 0.907·19-s − 0.441·20-s + 1.70·21-s + 0.424·22-s − 0.210·23-s + 1.44·24-s + 2.01·25-s + 0.661·26-s − 0.288·27-s − 0.324·28-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

 $$L(30)$$ $$\approx$$ $$2.61260$$ $$L(\frac12)$$ $$\approx$$ $$2.61260$$ $$L(\frac{61}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
good2 $$1 + 6.55e8T + 5.76e17T^{2}$$
3 $$1 - 1.58e14T + 1.41e28T^{2}$$
5 $$1 - 7.23e20T + 1.73e41T^{2}$$
7 $$1 - 1.08e25T + 7.25e49T^{2}$$
11 $$1 + 2.58e30T + 2.76e61T^{2}$$
13 $$1 + 5.56e32T + 5.28e65T^{2}$$
17 $$1 - 1.35e35T + 3.94e72T^{2}$$
19 $$1 - 4.79e37T + 2.79e75T^{2}$$
23 $$1 + 3.11e39T + 2.19e80T^{2}$$
29 $$1 - 1.44e43T + 1.91e86T^{2}$$
31 $$1 + 1.86e43T + 9.78e87T^{2}$$
37 $$1 - 1.60e46T + 3.34e92T^{2}$$
41 $$1 + 6.72e47T + 1.42e95T^{2}$$
43 $$1 - 1.54e48T + 2.36e96T^{2}$$
47 $$1 - 4.96e48T + 4.50e98T^{2}$$
53 $$1 - 3.89e50T + 5.39e101T^{2}$$
59 $$1 + 2.34e52T + 3.02e104T^{2}$$
61 $$1 + 7.62e51T + 2.16e105T^{2}$$
67 $$1 - 2.92e53T + 5.47e107T^{2}$$
71 $$1 - 2.90e54T + 1.67e109T^{2}$$
73 $$1 + 6.77e54T + 8.63e109T^{2}$$
79 $$1 + 1.07e56T + 9.12e111T^{2}$$
83 $$1 - 5.08e56T + 1.68e113T^{2}$$
89 $$1 - 3.31e57T + 1.03e115T^{2}$$
97 $$1 - 7.26e58T + 1.65e117T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−18.37118645506379680051992049775, −17.33229663480126584959774542782, −14.40088006701548876183634812640, −13.58164995328430937867782174192, −10.15477590426761803507527989292, −9.060582200406625685943185371625, −7.80551940878668052863160991095, −5.03166302061618668622875081168, −2.43010475932392204575042668200, −1.39111825076257501596238050121, 1.39111825076257501596238050121, 2.43010475932392204575042668200, 5.03166302061618668622875081168, 7.80551940878668052863160991095, 9.060582200406625685943185371625, 10.15477590426761803507527989292, 13.58164995328430937867782174192, 14.40088006701548876183634812640, 17.33229663480126584959774542782, 18.37118645506379680051992049775