# Properties

 Label 2-95-95.94-c4-0-28 Degree $2$ Conductor $95$ Sign $1$ Analytic cond. $9.82014$ Root an. cond. $3.13371$ Motivic weight $4$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 6.70·2-s + 4.47·3-s + 29.0·4-s + 25·5-s + 30.0·6-s + 87.2·8-s − 61·9-s + 167.·10-s − 62·11-s + 129.·12-s − 67.0·13-s + 111.·15-s + 121.·16-s − 409.·18-s + 361·19-s + 725.·20-s − 415.·22-s + 390.·24-s + 625·25-s − 450·26-s − 635.·27-s + 750.·30-s − 583.·32-s − 277.·33-s − 1.76e3·36-s − 2.64e3·37-s + 2.42e3·38-s + ⋯
 L(s)  = 1 + 1.67·2-s + 0.496·3-s + 1.81·4-s + 5-s + 0.833·6-s + 1.36·8-s − 0.753·9-s + 1.67·10-s − 0.512·11-s + 0.900·12-s − 0.396·13-s + 0.496·15-s + 0.472·16-s − 1.26·18-s + 19-s + 1.81·20-s − 0.859·22-s + 0.677·24-s + 25-s − 0.665·26-s − 0.871·27-s + 0.833·30-s − 0.569·32-s − 0.254·33-s − 1.36·36-s − 1.93·37-s + 1.67·38-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$95$$    =    $$5 \cdot 19$$ Sign: $1$ Analytic conductor: $$9.82014$$ Root analytic conductor: $$3.13371$$ Motivic weight: $$4$$ Rational: no Arithmetic: yes Character: $\chi_{95} (94, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 95,\ (\ :2),\ 1)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$5.154471411$$ $$L(\frac12)$$ $$\approx$$ $$5.154471411$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 - 25T$$
19 $$1 - 361T$$
good2 $$1 - 6.70T + 16T^{2}$$
3 $$1 - 4.47T + 81T^{2}$$
7 $$1 - 2.40e3T^{2}$$
11 $$1 + 62T + 1.46e4T^{2}$$
13 $$1 + 67.0T + 2.85e4T^{2}$$
17 $$1 - 8.35e4T^{2}$$
23 $$1 - 2.79e5T^{2}$$
29 $$1 - 7.07e5T^{2}$$
31 $$1 - 9.23e5T^{2}$$
37 $$1 + 2.64e3T + 1.87e6T^{2}$$
41 $$1 - 2.82e6T^{2}$$
43 $$1 - 3.41e6T^{2}$$
47 $$1 - 4.87e6T^{2}$$
53 $$1 - 791.T + 7.89e6T^{2}$$
59 $$1 - 1.21e7T^{2}$$
61 $$1 - 7.13e3T + 1.38e7T^{2}$$
67 $$1 + 6.07e3T + 2.01e7T^{2}$$
71 $$1 - 2.54e7T^{2}$$
73 $$1 - 2.83e7T^{2}$$
79 $$1 - 3.89e7T^{2}$$
83 $$1 - 4.74e7T^{2}$$
89 $$1 - 6.27e7T^{2}$$
97 $$1 + 1.54e4T + 8.85e7T^{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

−13.57334943631846293688007237545, −12.54643651824025999483925167147, −11.50513753515577700174771054630, −10.21498510190369114511038332773, −8.848879223948621224497156625464, −7.19133250618896566611253659942, −5.83737890371271249716412607421, −5.06408842290981846085975709017, −3.32073381822115920946437634512, −2.25572542021785236939160522503, 2.25572542021785236939160522503, 3.32073381822115920946437634512, 5.06408842290981846085975709017, 5.83737890371271249716412607421, 7.19133250618896566611253659942, 8.848879223948621224497156625464, 10.21498510190369114511038332773, 11.50513753515577700174771054630, 12.54643651824025999483925167147, 13.57334943631846293688007237545