Properties

Label 2-95-95.94-c4-0-15
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 $0$

Origins

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Normalization:  

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

\(L(\frac{5}{2})\) \(\approx\) \(0.6874299060\)
\(L(\frac12)\) \(\approx\) \(0.6874299060\)
\(L(3)\) 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

−13.18836900876708261982688968689, −11.67142682040378487914843403625, −10.82477619658826124356223081458, −9.898354266982772451834703022771, −9.007168983878526367304391043211, −7.911359142333311285752965927687, −6.55050977467251966660474252264, −5.44855585979411610412198506732, −2.53597628261584386114806763171, −0.882534227291798518873035730630, 0.882534227291798518873035730630, 2.53597628261584386114806763171, 5.44855585979411610412198506732, 6.55050977467251966660474252264, 7.911359142333311285752965927687, 9.007168983878526367304391043211, 9.898354266982772451834703022771, 10.82477619658826124356223081458, 11.67142682040378487914843403625, 13.18836900876708261982688968689

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