# Properties

 Degree $2$ Conductor $2$ Sign $1$ Motivic weight $49$ Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 1.67e7·2-s − 6.77e11·3-s + 2.81e14·4-s − 2.33e17·5-s − 1.13e19·6-s − 5.15e20·7-s + 4.72e21·8-s + 2.19e23·9-s − 3.90e24·10-s − 5.21e25·11-s − 1.90e26·12-s + 1.37e27·13-s − 8.64e27·14-s + 1.57e29·15-s + 7.92e28·16-s − 9.93e29·17-s + 3.67e30·18-s + 1.85e31·19-s − 6.55e31·20-s + 3.49e32·21-s − 8.74e32·22-s − 2.69e33·23-s − 3.19e33·24-s + 3.65e34·25-s + 2.30e34·26-s + 1.35e34·27-s − 1.45e35·28-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1.38·3-s + 0.5·4-s − 1.74·5-s − 0.978·6-s − 1.01·7-s + 0.353·8-s + 0.916·9-s − 1.23·10-s − 1.59·11-s − 0.692·12-s + 0.702·13-s − 0.719·14-s + 2.42·15-s + 0.250·16-s − 0.709·17-s + 0.647·18-s + 0.868·19-s − 0.874·20-s + 1.40·21-s − 1.12·22-s − 1.16·23-s − 0.489·24-s + 2.05·25-s + 0.497·26-s + 0.116·27-s − 0.508·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(50-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+49/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2$$ Sign: $1$ Motivic weight: $$49$$ Character: $\chi_{2} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2,\ (\ :49/2),\ 1)$$

## Particular Values

 $$L(25)$$ $$\approx$$ $$0.4296860814$$ $$L(\frac12)$$ $$\approx$$ $$0.4296860814$$ $$L(\frac{51}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 1.67e7T$$
good3 $$1 + 6.77e11T + 2.39e23T^{2}$$
5 $$1 + 2.33e17T + 1.77e34T^{2}$$
7 $$1 + 5.15e20T + 2.56e41T^{2}$$
11 $$1 + 5.21e25T + 1.06e51T^{2}$$
13 $$1 - 1.37e27T + 3.83e54T^{2}$$
17 $$1 + 9.93e29T + 1.95e60T^{2}$$
19 $$1 - 1.85e31T + 4.55e62T^{2}$$
23 $$1 + 2.69e33T + 5.30e66T^{2}$$
29 $$1 + 3.88e34T + 4.54e71T^{2}$$
31 $$1 + 8.58e35T + 1.19e73T^{2}$$
37 $$1 - 2.50e38T + 6.94e76T^{2}$$
41 $$1 - 2.12e38T + 1.06e79T^{2}$$
43 $$1 + 3.48e39T + 1.09e80T^{2}$$
47 $$1 - 2.11e40T + 8.56e81T^{2}$$
53 $$1 + 2.52e42T + 3.08e84T^{2}$$
59 $$1 + 4.29e43T + 5.91e86T^{2}$$
61 $$1 + 1.42e43T + 3.02e87T^{2}$$
67 $$1 - 2.95e44T + 3.00e89T^{2}$$
71 $$1 + 2.56e45T + 5.14e90T^{2}$$
73 $$1 + 5.17e45T + 2.00e91T^{2}$$
79 $$1 - 1.28e46T + 9.63e92T^{2}$$
83 $$1 - 8.58e46T + 1.08e94T^{2}$$
89 $$1 - 6.95e47T + 3.31e95T^{2}$$
97 $$1 - 7.53e48T + 2.24e97T^{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

−16.10783579996205814401263107504, −15.76584581557559534952040755148, −12.94587927038859581927827316841, −11.78308669141420651321467337635, −10.71265707610207407597099757239, −7.66940722106239554714971244108, −6.14815329312843700095546581612, −4.66930669446981506766037510014, −3.24898992284991690458622643024, −0.37534543783491370376268612734, 0.37534543783491370376268612734, 3.24898992284991690458622643024, 4.66930669446981506766037510014, 6.14815329312843700095546581612, 7.66940722106239554714971244108, 10.71265707610207407597099757239, 11.78308669141420651321467337635, 12.94587927038859581927827316841, 15.76584581557559534952040755148, 16.10783579996205814401263107504