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Properties

Label 2-95-19.11-c1-0-6
Degree 22
Conductor 9595
Sign 0.244+0.969i0.244 + 0.969i
Analytic cond. 0.7585780.758578
Root an. cond. 0.8709640.870964
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(95s/2 ΓC(s) L(s)=((0.244+0.969i) Λ‾(2−s)\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}
Λ(s)=(95s/2 ΓC(s+1/2) L(s)=((0.244+0.969i) Λ‾(1−s)\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: 22
Conductor: 9595    =    5⋅195 \cdot 19
Sign: 0.244+0.969i0.244 + 0.969i
Analytic conductor: 0.7585780.758578
Root analytic conductor: 0.8709640.870964
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ95(11,⋅)\chi_{95} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 95, ( :1/2), 0.244+0.969i)(2,\ 95,\ (\ :1/2),\ 0.244 + 0.969i)

Particular Values

L(1)L(1) ≈\approx 1.05943−0.825342i1.05943 - 0.825342i
L(12)L(\frac12) ≈\approx 1.05943−0.825342i1.05943 - 0.825342i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
19 1+(−0.149−4.35i)T 1 + (-0.149 - 4.35i)T
good2 1+(−0.832+1.44i)T+(−1−1.73i)T2 1 + (-0.832 + 1.44i)T + (-1 - 1.73i)T^{2}
3 1+(−0.579+1.00i)T+(−1.5−2.59i)T2 1 + (-0.579 + 1.00i)T + (-1.5 - 2.59i)T^{2}
7 1+2.43T+7T2 1 + 2.43T + 7T^{2}
11 1+5.75T+11T2 1 + 5.75T + 11T^{2}
13 1+(−0.797−1.38i)T+(−6.5+11.2i)T2 1 + (-0.797 - 1.38i)T + (-6.5 + 11.2i)T^{2}
17 1+(−2.99+5.18i)T+(−8.5−14.7i)T2 1 + (-2.99 + 5.18i)T + (-8.5 - 14.7i)T^{2}
23 1+(−0.470−0.814i)T+(−11.5+19.9i)T2 1 + (-0.470 - 0.814i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.30+2.26i)T+(−14.5+25.1i)T2 1 + (1.30 + 2.26i)T + (-14.5 + 25.1i)T^{2}
31 1+5.26T+31T2 1 + 5.26T + 31T^{2}
37 1+2.89T+37T2 1 + 2.89T + 37T^{2}
41 1+(−3.15+5.46i)T+(−20.5−35.5i)T2 1 + (-3.15 + 5.46i)T + (-20.5 - 35.5i)T^{2}
43 1+(2.26−3.93i)T+(−21.5−37.2i)T2 1 + (2.26 - 3.93i)T + (-21.5 - 37.2i)T^{2}
47 1+(4.47+7.75i)T+(−23.5+40.7i)T2 1 + (4.47 + 7.75i)T + (-23.5 + 40.7i)T^{2}
53 1+(−1.09−1.90i)T+(−26.5+45.8i)T2 1 + (-1.09 - 1.90i)T + (-26.5 + 45.8i)T^{2}
59 1+(−5.39+9.35i)T+(−29.5−51.0i)T2 1 + (-5.39 + 9.35i)T + (-29.5 - 51.0i)T^{2}
61 1+(−5.26−9.11i)T+(−30.5+52.8i)T2 1 + (-5.26 - 9.11i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.504+0.874i)T+(−33.5+58.0i)T2 1 + (0.504 + 0.874i)T + (-33.5 + 58.0i)T^{2}
71 1+(4.41−7.65i)T+(−35.5−61.4i)T2 1 + (4.41 - 7.65i)T + (-35.5 - 61.4i)T^{2}
73 1+(−5.12+8.87i)T+(−36.5−63.2i)T2 1 + (-5.12 + 8.87i)T + (-36.5 - 63.2i)T^{2}
79 1+(3.80−6.58i)T+(−39.5−68.4i)T2 1 + (3.80 - 6.58i)T + (-39.5 - 68.4i)T^{2}
83 1−3.11T+83T2 1 - 3.11T + 83T^{2}
89 1+(−5.55−9.62i)T+(−44.5+77.0i)T2 1 + (-5.55 - 9.62i)T + (-44.5 + 77.0i)T^{2}
97 1+(2.02−3.51i)T+(−48.5−84.0i)T2 1 + (2.02 - 3.51i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

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

Graph of the ZZ-function along the critical line