# Properties

 Label 1-95-95.94-r1-0-0 Degree $1$ Conductor $95$ Sign $1$ Analytic cond. $10.2091$ Root an. cond. $10.2091$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 11-s + 12-s + 13-s − 14-s + 16-s − 17-s + 18-s − 21-s + 22-s − 23-s + 24-s + 26-s + 27-s − 28-s − 29-s − 31-s + 32-s + 33-s − 34-s + 36-s + ⋯
 L(s)  = 1 + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 11-s + 12-s + 13-s − 14-s + 16-s − 17-s + 18-s − 21-s + 22-s − 23-s + 24-s + 26-s + 27-s − 28-s − 29-s − 31-s + 32-s + 33-s − 34-s + 36-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$1$$ Conductor: $$95$$    =    $$5 \cdot 19$$ Sign: $1$ Analytic conductor: $$10.2091$$ Root analytic conductor: $$10.2091$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{95} (94, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(1,\ 95,\ (1:\ ),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$4.253930769$$ $$L(\frac12)$$ $$\approx$$ $$4.253930769$$ $$L(1)$$ $$\approx$$ $$2.578564842$$ $$L(1)$$ $$\approx$$ $$2.578564842$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
19 $$1$$
good2 $$1 + T$$
3 $$1 + T$$
7 $$1 - T$$
11 $$1 + T$$
13 $$1 + T$$
17 $$1 - T$$
23 $$1 - T$$
29 $$1 - T$$
31 $$1 - T$$
37 $$1 + T$$
41 $$1 - T$$
43 $$1 - T$$
47 $$1 - T$$
53 $$1 + T$$
59 $$1 - T$$
61 $$1 + T$$
67 $$1 + T$$
71 $$1 - T$$
73 $$1 - T$$
79 $$1 - T$$
83 $$1 - T$$
89 $$1 - T$$
97 $$1 + T$$
show less
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

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

−30.23265183696012695184083300872, −29.27938403823280748967589108167, −28.00054946578319882160940361777, −26.39784457802140517588358518996, −25.58214084168719201698024169407, −24.77561474289418773080537418449, −23.68109520822372539038382383830, −22.4085010324865502905692068217, −21.66226644999998078812797091575, −20.23818906997674775202237395394, −19.82328799751565574456269375641, −18.535448014834162256039685359257, −16.565774884582515894149541814869, −15.657278846743730856281961000028, −14.6451016714692529814686719904, −13.536157210440015496201160629741, −12.87438993327497290816145860958, −11.447319410849700177967245509866, −9.94526501369435831524904146612, −8.67695243363343554902904315633, −7.09458909151310291308741027052, −6.11608180143257073848335422001, −4.14550362368825989000989851231, −3.34640088272081680339402511009, −1.83333896167925940476672330751, 1.83333896167925940476672330751, 3.34640088272081680339402511009, 4.14550362368825989000989851231, 6.11608180143257073848335422001, 7.09458909151310291308741027052, 8.67695243363343554902904315633, 9.94526501369435831524904146612, 11.447319410849700177967245509866, 12.87438993327497290816145860958, 13.536157210440015496201160629741, 14.6451016714692529814686719904, 15.657278846743730856281961000028, 16.565774884582515894149541814869, 18.535448014834162256039685359257, 19.82328799751565574456269375641, 20.23818906997674775202237395394, 21.66226644999998078812797091575, 22.4085010324865502905692068217, 23.68109520822372539038382383830, 24.77561474289418773080537418449, 25.58214084168719201698024169407, 26.39784457802140517588358518996, 28.00054946578319882160940361777, 29.27938403823280748967589108167, 30.23265183696012695184083300872