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Properties

Label 2-171-19.11-c1-0-7
Degree 22
Conductor 171171
Sign −0.567+0.823i-0.567 + 0.823i
Analytic cond. 1.365441.36544
Root an. cond. 1.168521.16852
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  + (1.16 − 2.02i)2-s + (−1.72 − 2.98i)4-s + (0.524 − 0.908i)5-s − 3.44·7-s − 3.38·8-s + (−1.22 − 2.12i)10-s + 5.71·11-s + (0.5 + 0.866i)13-s + (−4.02 + 6.97i)14-s + (−0.500 + 0.866i)16-s + (−1.04 + 1.81i)17-s + (1 + 4.24i)19-s − 3.61·20-s + (6.67 − 11.5i)22-s + (−1.80 − 3.13i)23-s + ⋯
L(s)  = 1  + (0.825 − 1.42i)2-s + (−0.862 − 1.49i)4-s + (0.234 − 0.406i)5-s − 1.30·7-s − 1.19·8-s + (−0.387 − 0.670i)10-s + 1.72·11-s + (0.138 + 0.240i)13-s + (−1.07 + 1.86i)14-s + (−0.125 + 0.216i)16-s + (−0.254 + 0.440i)17-s + (0.229 + 0.973i)19-s − 0.809·20-s + (1.42 − 2.46i)22-s + (−0.377 − 0.653i)23-s + ⋯

Functional equation

Λ(s)=(171s/2 ΓC(s) L(s)=((−0.567+0.823i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(171s/2 ΓC(s+1/2) L(s)=((−0.567+0.823i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 171171    =    32⋅193^{2} \cdot 19
Sign: −0.567+0.823i-0.567 + 0.823i
Analytic conductor: 1.365441.36544
Root analytic conductor: 1.168521.16852
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ171(163,⋅)\chi_{171} (163, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 171, ( :1/2), −0.567+0.823i)(2,\ 171,\ (\ :1/2),\ -0.567 + 0.823i)

Particular Values

L(1)L(1) ≈\approx 0.753362−1.43317i0.753362 - 1.43317i
L(12)L(\frac12) ≈\approx 0.753362−1.43317i0.753362 - 1.43317i
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)
bad3 1 1
19 1+(−1−4.24i)T 1 + (-1 - 4.24i)T
good2 1+(−1.16+2.02i)T+(−1−1.73i)T2 1 + (-1.16 + 2.02i)T + (-1 - 1.73i)T^{2}
5 1+(−0.524+0.908i)T+(−2.5−4.33i)T2 1 + (-0.524 + 0.908i)T + (-2.5 - 4.33i)T^{2}
7 1+3.44T+7T2 1 + 3.44T + 7T^{2}
11 1−5.71T+11T2 1 - 5.71T + 11T^{2}
13 1+(−0.5−0.866i)T+(−6.5+11.2i)T2 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2}
17 1+(1.04−1.81i)T+(−8.5−14.7i)T2 1 + (1.04 - 1.81i)T + (-8.5 - 14.7i)T^{2}
23 1+(1.80+3.13i)T+(−11.5+19.9i)T2 1 + (1.80 + 3.13i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3.61−6.26i)T+(−14.5+25.1i)T2 1 + (-3.61 - 6.26i)T + (-14.5 + 25.1i)T^{2}
31 1+9.44T+31T2 1 + 9.44T + 31T^{2}
37 1−3.89T+37T2 1 - 3.89T + 37T^{2}
41 1+(−4.66+8.08i)T+(−20.5−35.5i)T2 1 + (-4.66 + 8.08i)T + (-20.5 - 35.5i)T^{2}
43 1+(3.17−5.49i)T+(−21.5−37.2i)T2 1 + (3.17 - 5.49i)T + (-21.5 - 37.2i)T^{2}
47 1+(4.66+8.08i)T+(−23.5+40.7i)T2 1 + (4.66 + 8.08i)T + (-23.5 + 40.7i)T^{2}
53 1+(0.524+0.908i)T+(−26.5+45.8i)T2 1 + (0.524 + 0.908i)T + (-26.5 + 45.8i)T^{2}
59 1+(3.90−6.76i)T+(−29.5−51.0i)T2 1 + (3.90 - 6.76i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.5+4.33i)T+(−30.5+52.8i)T2 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.174+0.301i)T+(−33.5+58.0i)T2 1 + (0.174 + 0.301i)T + (-33.5 + 58.0i)T^{2}
71 1+(3.61−6.26i)T+(−35.5−61.4i)T2 1 + (3.61 - 6.26i)T + (-35.5 - 61.4i)T^{2}
73 1+(2.5−4.33i)T+(−36.5−63.2i)T2 1 + (2.5 - 4.33i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.174−0.301i)T+(−39.5−68.4i)T2 1 + (0.174 - 0.301i)T + (-39.5 - 68.4i)T^{2}
83 1+11.4T+83T2 1 + 11.4T + 83T^{2}
89 1+(−2.62−4.54i)T+(−44.5+77.0i)T2 1 + (-2.62 - 4.54i)T + (-44.5 + 77.0i)T^{2}
97 1+(1.55−2.68i)T+(−48.5−84.0i)T2 1 + (1.55 - 2.68i)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

−12.53330189331524522852728494892, −11.66587319075976444966662476591, −10.58775125566395322707918686422, −9.610582587923904372606763061310, −8.929230373849495247194570113950, −6.79775061315552148733203431657, −5.69932283889306553982380002765, −4.15500534839769717008787518503, −3.33908773496856145759819457692, −1.55527538572307297230851856321, 3.27209989116301653810870929278, 4.45028810441098974392498080797, 6.05774345936676534629685546238, 6.51947197923179258827423026222, 7.44647373838757797092132209397, 8.944538258651007672583144140959, 9.763344036572334146425231920320, 11.36140530109858236375060029902, 12.54798826804145465692591500639, 13.40140601187099439317413135446

Graph of the ZZ-function along the critical line