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Properties

Label 2-475-19.11-c1-0-24
Degree 22
Conductor 475475
Sign −0.634+0.772i-0.634 + 0.772i
Analytic cond. 3.792893.79289
Root an. cond. 1.947531.94753
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  + (−0.740 + 1.28i)2-s + (1.42 − 2.47i)3-s + (−0.0969 − 0.167i)4-s + (2.11 + 3.66i)6-s − 3.78·7-s − 2.67·8-s + (−2.58 − 4.46i)9-s − 5.59·11-s − 0.554·12-s + (−2.45 − 4.24i)13-s + (2.80 − 4.85i)14-s + (2.17 − 3.76i)16-s + (0.875 − 1.51i)17-s + 7.64·18-s + (0.636 + 4.31i)19-s + ⋯
L(s)  = 1  + (−0.523 + 0.907i)2-s + (0.824 − 1.42i)3-s + (−0.0484 − 0.0839i)4-s + (0.863 + 1.49i)6-s − 1.43·7-s − 0.945·8-s + (−0.860 − 1.48i)9-s − 1.68·11-s − 0.159·12-s + (−0.680 − 1.17i)13-s + (0.749 − 1.29i)14-s + (0.543 − 0.941i)16-s + (0.212 − 0.367i)17-s + 1.80·18-s + (0.145 + 0.989i)19-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=((−0.634+0.772i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=((−0.634+0.772i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: −0.634+0.772i-0.634 + 0.772i
Analytic conductor: 3.792893.79289
Root analytic conductor: 1.947531.94753
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ475(201,⋅)\chi_{475} (201, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 475, ( :1/2), −0.634+0.772i)(2,\ 475,\ (\ :1/2),\ -0.634 + 0.772i)

Particular Values

L(1)L(1) ≈\approx 0.194282−0.411134i0.194282 - 0.411134i
L(12)L(\frac12) ≈\approx 0.194282−0.411134i0.194282 - 0.411134i
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 1
19 1+(−0.636−4.31i)T 1 + (-0.636 - 4.31i)T
good2 1+(0.740−1.28i)T+(−1−1.73i)T2 1 + (0.740 - 1.28i)T + (-1 - 1.73i)T^{2}
3 1+(−1.42+2.47i)T+(−1.5−2.59i)T2 1 + (-1.42 + 2.47i)T + (-1.5 - 2.59i)T^{2}
7 1+3.78T+7T2 1 + 3.78T + 7T^{2}
11 1+5.59T+11T2 1 + 5.59T + 11T^{2}
13 1+(2.45+4.24i)T+(−6.5+11.2i)T2 1 + (2.45 + 4.24i)T + (-6.5 + 11.2i)T^{2}
17 1+(−0.875+1.51i)T+(−8.5−14.7i)T2 1 + (-0.875 + 1.51i)T + (-8.5 - 14.7i)T^{2}
23 1+(0.290+0.503i)T+(−11.5+19.9i)T2 1 + (0.290 + 0.503i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.832+1.44i)T+(−14.5+25.1i)T2 1 + (0.832 + 1.44i)T + (-14.5 + 25.1i)T^{2}
31 1−7.01T+31T2 1 - 7.01T + 31T^{2}
37 1−2.36T+37T2 1 - 2.36T + 37T^{2}
41 1+(0.417−0.723i)T+(−20.5−35.5i)T2 1 + (0.417 - 0.723i)T + (-20.5 - 35.5i)T^{2}
43 1+(−0.535+0.927i)T+(−21.5−37.2i)T2 1 + (-0.535 + 0.927i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.93+3.34i)T+(−23.5+40.7i)T2 1 + (1.93 + 3.34i)T + (-23.5 + 40.7i)T^{2}
53 1+(−3.39−5.88i)T+(−26.5+45.8i)T2 1 + (-3.39 - 5.88i)T + (-26.5 + 45.8i)T^{2}
59 1+(−0.204+0.353i)T+(−29.5−51.0i)T2 1 + (-0.204 + 0.353i)T + (-29.5 - 51.0i)T^{2}
61 1+(6.98+12.0i)T+(−30.5+52.8i)T2 1 + (6.98 + 12.0i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.390+0.676i)T+(−33.5+58.0i)T2 1 + (0.390 + 0.676i)T + (-33.5 + 58.0i)T^{2}
71 1+(3.18−5.52i)T+(−35.5−61.4i)T2 1 + (3.18 - 5.52i)T + (-35.5 - 61.4i)T^{2}
73 1+(1.44−2.50i)T+(−36.5−63.2i)T2 1 + (1.44 - 2.50i)T + (-36.5 - 63.2i)T^{2}
79 1+(−6.25+10.8i)T+(−39.5−68.4i)T2 1 + (-6.25 + 10.8i)T + (-39.5 - 68.4i)T^{2}
83 1+10.3T+83T2 1 + 10.3T + 83T^{2}
89 1+(8.92+15.4i)T+(−44.5+77.0i)T2 1 + (8.92 + 15.4i)T + (-44.5 + 77.0i)T^{2}
97 1+(5.49−9.52i)T+(−48.5−84.0i)T2 1 + (5.49 - 9.52i)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

−10.27747347628217859737665544481, −9.583953662078833275275935480545, −8.388688612982934047630345236692, −7.83847393112498884660742091195, −7.29649252212206789852541178795, −6.36398132602313013751205238392, −5.56377324253329195883857251919, −3.10766019931239889004477538595, −2.67149044086321058488672341534, −0.26612347484237768736870817335, 2.56478735110779149141995716498, 3.00761041254087793203745019266, 4.26905804726042009955768048174, 5.44298691920395577942770580314, 6.76818940291783515479764649996, 8.191416463859144086378384875021, 9.207501392571091436554399812577, 9.629086925121742419757190989992, 10.26069167462378701618308759629, 10.85719300579328434375940571406

Graph of the ZZ-function along the critical line