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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

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Normalization:  

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 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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

Graph of the $Z$-function along the critical line