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Properties

Label 2-95-19.7-c1-0-3
Degree $2$
Conductor $95$
Sign $0.244 - 0.969i$
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  + (0.832 + 1.44i)2-s + (0.579 + 1.00i)3-s + (−0.385 + 0.667i)4-s + (−0.5 − 0.866i)5-s + (−0.965 + 1.67i)6-s − 2.43·7-s + 2.04·8-s + (0.827 − 1.43i)9-s + (0.832 − 1.44i)10-s − 5.75·11-s − 0.893·12-s + (0.797 − 1.38i)13-s + (−2.02 − 3.51i)14-s + (0.579 − 1.00i)15-s + (2.47 + 4.28i)16-s + (2.99 + 5.18i)17-s + ⋯
L(s)  = 1  + (0.588 + 1.01i)2-s + (0.334 + 0.579i)3-s + (−0.192 + 0.333i)4-s + (−0.223 − 0.387i)5-s + (−0.394 + 0.682i)6-s − 0.920·7-s + 0.723·8-s + (0.275 − 0.477i)9-s + (0.263 − 0.455i)10-s − 1.73·11-s − 0.258·12-s + (0.221 − 0.383i)13-s + (−0.541 − 0.938i)14-s + (0.149 − 0.259i)15-s + (0.618 + 1.07i)16-s + (0.725 + 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.244 - 0.969i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.244 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05943 + 0.825342i\)
\(L(\frac12)\) \(\approx\) \(1.05943 + 0.825342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.149 + 4.35i)T \)
good2 \( 1 + (-0.832 - 1.44i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.579 - 1.00i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 2.43T + 7T^{2} \)
11 \( 1 + 5.75T + 11T^{2} \)
13 \( 1 + (-0.797 + 1.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.99 - 5.18i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.470 + 0.814i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.30 - 2.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 + 2.89T + 37T^{2} \)
41 \( 1 + (-3.15 - 5.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.26 + 3.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.47 - 7.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.09 + 1.90i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.39 - 9.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.26 + 9.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.504 - 0.874i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.41 + 7.65i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.12 - 8.87i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.80 + 6.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.11T + 83T^{2} \)
89 \( 1 + (-5.55 + 9.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.02 + 3.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.54874534009600698051124125936, −13.05637004340446750175659203340, −12.82337044684863547482026097445, −10.74531488682028142385503912449, −9.896683940117095071802750474267, −8.485288031526507491745496813211, −7.33746146797625876524125585321, −6.01627479357606074512520245157, −4.88222841823786374027609550257, −3.43145695284455644466045054413, 2.34737899186682024585389260596, 3.46472210352320493045315886827, 5.24124426899623972660220834703, 7.15562095679816148750554807057, 7.942396577047318941995325308265, 9.879168573822033094588934840321, 10.66642836378644564513874016978, 11.85561052462833769707490500790, 12.86768598469082587225251727402, 13.37415616752082970205865156648

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