# Properties

 Degree $2$ Conductor $6025$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 2.70·2-s + 2.50·3-s + 5.29·4-s − 6.77·6-s − 0.354·7-s − 8.89·8-s + 3.29·9-s + 4.18·11-s + 13.2·12-s + 3.72·13-s + 0.958·14-s + 13.4·16-s + 6.46·17-s − 8.88·18-s − 1.31·19-s − 0.890·21-s − 11.3·22-s + 4.10·23-s − 22.3·24-s − 10.0·26-s + 0.728·27-s − 1.87·28-s − 8.85·29-s + 5.11·31-s − 18.4·32-s + 10.5·33-s − 17.4·34-s + ⋯
 L(s)  = 1 − 1.90·2-s + 1.44·3-s + 2.64·4-s − 2.76·6-s − 0.134·7-s − 3.14·8-s + 1.09·9-s + 1.26·11-s + 3.83·12-s + 1.03·13-s + 0.256·14-s + 3.35·16-s + 1.56·17-s − 2.09·18-s − 0.300·19-s − 0.194·21-s − 2.41·22-s + 0.856·23-s − 4.55·24-s − 1.97·26-s + 0.140·27-s − 0.355·28-s − 1.64·29-s + 0.918·31-s − 3.26·32-s + 1.82·33-s − 2.99·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$6025$$    =    $$5^{2} \cdot 241$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{6025} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 6025,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.820061803$$ $$L(\frac12)$$ $$\approx$$ $$1.820061803$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
241 $$1 - T$$
good2 $$1 + 2.70T + 2T^{2}$$
3 $$1 - 2.50T + 3T^{2}$$
7 $$1 + 0.354T + 7T^{2}$$
11 $$1 - 4.18T + 11T^{2}$$
13 $$1 - 3.72T + 13T^{2}$$
17 $$1 - 6.46T + 17T^{2}$$
19 $$1 + 1.31T + 19T^{2}$$
23 $$1 - 4.10T + 23T^{2}$$
29 $$1 + 8.85T + 29T^{2}$$
31 $$1 - 5.11T + 31T^{2}$$
37 $$1 + 5.41T + 37T^{2}$$
41 $$1 - 11.8T + 41T^{2}$$
43 $$1 + 0.673T + 43T^{2}$$
47 $$1 + 5.22T + 47T^{2}$$
53 $$1 - 9.92T + 53T^{2}$$
59 $$1 + 1.23T + 59T^{2}$$
61 $$1 - 4.04T + 61T^{2}$$
67 $$1 - 14.0T + 67T^{2}$$
71 $$1 - 13.0T + 71T^{2}$$
73 $$1 + 7.76T + 73T^{2}$$
79 $$1 + 1.17T + 79T^{2}$$
83 $$1 + 6.25T + 83T^{2}$$
89 $$1 - 3.80T + 89T^{2}$$
97 $$1 - 9.91T + 97T^{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

−8.179763805116092928151195459172, −7.76458729862475505946014067873, −7.02077981730225841302149636073, −6.40566406270077212170620869456, −5.56310808258554733405527658971, −3.83074232210901338267723503995, −3.40044102133817410058150930629, −2.51064488947241900811206141122, −1.60136278667572095260678791810, −0.963880784943121677122008610201, 0.963880784943121677122008610201, 1.60136278667572095260678791810, 2.51064488947241900811206141122, 3.40044102133817410058150930629, 3.83074232210901338267723503995, 5.56310808258554733405527658971, 6.40566406270077212170620869456, 7.02077981730225841302149636073, 7.76458729862475505946014067873, 8.179763805116092928151195459172