# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.49·2-s − 3.12·3-s + 0.226·4-s − 4.66·6-s − 4.66·7-s − 2.64·8-s + 6.77·9-s − 5.92·11-s − 0.707·12-s + 4.29·13-s − 6.95·14-s − 4.40·16-s + 5.72·17-s + 10.1·18-s − 1.23·19-s + 14.5·21-s − 8.83·22-s + 3.63·23-s + 8.27·24-s + 6.40·26-s − 11.8·27-s − 1.05·28-s + 1.81·29-s − 1.11·31-s − 1.27·32-s + 18.5·33-s + 8.54·34-s + ⋯
 L(s)  = 1 + 1.05·2-s − 1.80·3-s + 0.113·4-s − 1.90·6-s − 1.76·7-s − 0.935·8-s + 2.25·9-s − 1.78·11-s − 0.204·12-s + 1.19·13-s − 1.85·14-s − 1.10·16-s + 1.38·17-s + 2.38·18-s − 0.282·19-s + 3.17·21-s − 1.88·22-s + 0.757·23-s + 1.68·24-s + 1.25·26-s − 2.27·27-s − 0.199·28-s + 0.336·29-s − 0.200·31-s − 0.225·32-s + 3.22·33-s + 1.46·34-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: $$1$$ Selberg data: $$(2,\ 6025,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 - 1.49T + 2T^{2}$$
3 $$1 + 3.12T + 3T^{2}$$
7 $$1 + 4.66T + 7T^{2}$$
11 $$1 + 5.92T + 11T^{2}$$
13 $$1 - 4.29T + 13T^{2}$$
17 $$1 - 5.72T + 17T^{2}$$
19 $$1 + 1.23T + 19T^{2}$$
23 $$1 - 3.63T + 23T^{2}$$
29 $$1 - 1.81T + 29T^{2}$$
31 $$1 + 1.11T + 31T^{2}$$
37 $$1 - 7.32T + 37T^{2}$$
41 $$1 + 6.45T + 41T^{2}$$
43 $$1 - 4.57T + 43T^{2}$$
47 $$1 - 7.71T + 47T^{2}$$
53 $$1 + 10.1T + 53T^{2}$$
59 $$1 + 5.66T + 59T^{2}$$
61 $$1 - 4.76T + 61T^{2}$$
67 $$1 - 14.0T + 67T^{2}$$
71 $$1 + 3.96T + 71T^{2}$$
73 $$1 + 8.92T + 73T^{2}$$
79 $$1 - 1.77T + 79T^{2}$$
83 $$1 - 9.93T + 83T^{2}$$
89 $$1 + 7.03T + 89T^{2}$$
97 $$1 + 5.03T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$