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Properties

Label 2-95-95.37-c1-0-3
Degree $2$
Conductor $95$
Sign $0.125 - 0.992i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.813 + 0.813i)2-s + (2.01 + 2.01i)3-s + 0.675i·4-s + (1.48 − 1.67i)5-s − 3.28·6-s + (−2.67 − 2.67i)7-s + (−2.17 − 2.17i)8-s + 5.15i·9-s + (0.157 + 2.56i)10-s + 2.15·11-s + (−1.36 + 1.36i)12-s + (−2.01 − 2.01i)13-s + 4.35·14-s + (6.37 − 0.391i)15-s + 2.19·16-s + (1.80 + 1.80i)17-s + ⋯
L(s)  = 1  + (−0.575 + 0.575i)2-s + (1.16 + 1.16i)3-s + 0.337i·4-s + (0.662 − 0.749i)5-s − 1.34·6-s + (−1.01 − 1.01i)7-s + (−0.769 − 0.769i)8-s + 1.71i·9-s + (0.0499 + 0.812i)10-s + 0.650·11-s + (−0.393 + 0.393i)12-s + (−0.560 − 0.560i)13-s + 1.16·14-s + (1.64 − 0.101i)15-s + 0.548·16-s + (0.438 + 0.438i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.125 - 0.992i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.125 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771899 + 0.680324i\)
\(L(\frac12)\) \(\approx\) \(0.771899 + 0.680324i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.48 + 1.67i)T \)
19 \( 1 + (4.35 - 0.193i)T \)
good2 \( 1 + (0.813 - 0.813i)T - 2iT^{2} \)
3 \( 1 + (-2.01 - 2.01i)T + 3iT^{2} \)
7 \( 1 + (2.67 + 2.67i)T + 7iT^{2} \)
11 \( 1 - 2.15T + 11T^{2} \)
13 \( 1 + (2.01 + 2.01i)T + 13iT^{2} \)
17 \( 1 + (-1.80 - 1.80i)T + 17iT^{2} \)
23 \( 1 + (2.67 - 2.67i)T - 23iT^{2} \)
29 \( 1 - 7.29T + 29T^{2} \)
31 \( 1 - 2.09iT - 31T^{2} \)
37 \( 1 + (-0.391 + 0.391i)T - 37iT^{2} \)
41 \( 1 - 3.51iT - 41T^{2} \)
43 \( 1 + (5.24 - 5.24i)T - 43iT^{2} \)
47 \( 1 + (-1.63 - 1.63i)T + 47iT^{2} \)
53 \( 1 + (4.43 + 4.43i)T + 53iT^{2} \)
59 \( 1 - 3.51T + 59T^{2} \)
61 \( 1 + 4.93T + 61T^{2} \)
67 \( 1 + (7.68 - 7.68i)T - 67iT^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 + (-9.73 + 9.73i)T - 73iT^{2} \)
79 \( 1 + 5.19T + 79T^{2} \)
83 \( 1 + (8.44 - 8.44i)T - 83iT^{2} \)
89 \( 1 + 5.60T + 89T^{2} \)
97 \( 1 + (-5.59 + 5.59i)T - 97iT^{2} \)
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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

−14.37548186271660968449969030670, −13.37420247841187056781240608255, −12.45392606768375987222057661917, −10.20126286251546709548414321934, −9.763636453705239144269153590374, −8.813574104483761615913033159648, −7.922630953152926973195660816701, −6.41560051281070582033981036562, −4.37219894822569703828436081962, −3.23324643996492342027545112368, 2.05757274916126030233543167620, 2.88689140630869637094612465451, 6.08676141798314084360626763077, 6.86656599709386180605114161372, 8.570159026338109863550041945020, 9.324857521957119569819331983249, 10.16900088224664672843861404065, 11.82011312260060435911920092422, 12.62217652612726306235541305540, 13.91706903465476199499126264872

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