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Properties

Label 2-475-19.11-c1-0-18
Degree 22
Conductor 475475
Sign 0.983+0.178i0.983 + 0.178i
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  + (−0.155 + 0.269i)2-s + (1.12 − 1.94i)3-s + (0.951 + 1.64i)4-s + (0.349 + 0.604i)6-s + 3.96·7-s − 1.21·8-s + (−1.01 − 1.76i)9-s + 0.361·11-s + 4.27·12-s + (−1.25 − 2.17i)13-s + (−0.617 + 1.06i)14-s + (−1.71 + 2.96i)16-s + (−0.00464 + 0.00803i)17-s + 0.633·18-s + (−3.45 + 2.65i)19-s + ⋯
L(s)  = 1  + (−0.109 + 0.190i)2-s + (0.647 − 1.12i)3-s + (0.475 + 0.824i)4-s + (0.142 + 0.246i)6-s + 1.50·7-s − 0.429·8-s + (−0.339 − 0.587i)9-s + 0.108·11-s + 1.23·12-s + (−0.348 − 0.604i)13-s + (−0.165 + 0.285i)14-s + (−0.428 + 0.742i)16-s + (−0.00112 + 0.00194i)17-s + 0.149·18-s + (−0.793 + 0.609i)19-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=((0.983+0.178i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=((0.983+0.178i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: 0.983+0.178i0.983 + 0.178i
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), 0.983+0.178i)(2,\ 475,\ (\ :1/2),\ 0.983 + 0.178i)

Particular Values

L(1)L(1) ≈\approx 2.01350−0.181033i2.01350 - 0.181033i
L(12)L(\frac12) ≈\approx 2.01350−0.181033i2.01350 - 0.181033i
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+(3.45−2.65i)T 1 + (3.45 - 2.65i)T
good2 1+(0.155−0.269i)T+(−1−1.73i)T2 1 + (0.155 - 0.269i)T + (-1 - 1.73i)T^{2}
3 1+(−1.12+1.94i)T+(−1.5−2.59i)T2 1 + (-1.12 + 1.94i)T + (-1.5 - 2.59i)T^{2}
7 1−3.96T+7T2 1 - 3.96T + 7T^{2}
11 1−0.361T+11T2 1 - 0.361T + 11T^{2}
13 1+(1.25+2.17i)T+(−6.5+11.2i)T2 1 + (1.25 + 2.17i)T + (-6.5 + 11.2i)T^{2}
17 1+(0.00464−0.00803i)T+(−8.5−14.7i)T2 1 + (0.00464 - 0.00803i)T + (-8.5 - 14.7i)T^{2}
23 1+(−2.70−4.68i)T+(−11.5+19.9i)T2 1 + (-2.70 - 4.68i)T + (-11.5 + 19.9i)T^{2}
29 1+(4.72+8.18i)T+(−14.5+25.1i)T2 1 + (4.72 + 8.18i)T + (-14.5 + 25.1i)T^{2}
31 1−3.66T+31T2 1 - 3.66T + 31T^{2}
37 1−0.0596T+37T2 1 - 0.0596T + 37T^{2}
41 1+(1.85−3.21i)T+(−20.5−35.5i)T2 1 + (1.85 - 3.21i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2.10+3.64i)T+(−21.5−37.2i)T2 1 + (-2.10 + 3.64i)T + (-21.5 - 37.2i)T^{2}
47 1+(6.45+11.1i)T+(−23.5+40.7i)T2 1 + (6.45 + 11.1i)T + (-23.5 + 40.7i)T^{2}
53 1+(5.48+9.50i)T+(−26.5+45.8i)T2 1 + (5.48 + 9.50i)T + (-26.5 + 45.8i)T^{2}
59 1+(−2.65+4.60i)T+(−29.5−51.0i)T2 1 + (-2.65 + 4.60i)T + (-29.5 - 51.0i)T^{2}
61 1+(−4.44−7.70i)T+(−30.5+52.8i)T2 1 + (-4.44 - 7.70i)T + (-30.5 + 52.8i)T^{2}
67 1+(−2.32−4.02i)T+(−33.5+58.0i)T2 1 + (-2.32 - 4.02i)T + (-33.5 + 58.0i)T^{2}
71 1+(7.68−13.3i)T+(−35.5−61.4i)T2 1 + (7.68 - 13.3i)T + (-35.5 - 61.4i)T^{2}
73 1+(4.83−8.36i)T+(−36.5−63.2i)T2 1 + (4.83 - 8.36i)T + (-36.5 - 63.2i)T^{2}
79 1+(6.70−11.6i)T+(−39.5−68.4i)T2 1 + (6.70 - 11.6i)T + (-39.5 - 68.4i)T^{2}
83 1+15.5T+83T2 1 + 15.5T + 83T^{2}
89 1+(−2.08−3.61i)T+(−44.5+77.0i)T2 1 + (-2.08 - 3.61i)T + (-44.5 + 77.0i)T^{2}
97 1+(−1.87+3.24i)T+(−48.5−84.0i)T2 1 + (-1.87 + 3.24i)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

−11.42083523459798674206609504175, −10.08286341528978993546592449844, −8.577676998668760482851670383408, −8.157769479605353385178716237274, −7.53849097682163694850012135350, −6.75146875870722341184452330710, −5.43490807777781566439265872357, −3.99526827464811868725654850505, −2.58087012120811510063666994177, −1.66171642933155908020893713013, 1.64745820740410844217129635511, 2.89564707018477251374468674271, 4.51965393208119485609087909429, 4.90567731615423234223719183663, 6.33705869592074149550969234813, 7.48979234568587720470428849406, 8.750844420032352301168763966750, 9.174608944156344960858333621352, 10.27363624089657501190039413962, 10.89308570719035888725585007674

Graph of the ZZ-function along the critical line