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Properties

Label 2-171-19.11-c1-0-0
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 0.105361−0.200437i0.105361 - 0.200437i
L(12)L(\frac12) ≈\approx 0.105361−0.200437i0.105361 - 0.200437i
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

−13.52469750368033338418998966224, −12.72262785535506406658835968842, −11.11468805809149319121343795866, −9.927738319988938078742900261705, −9.374323007642214723034162031078, −7.945723517438357304902467071890, −7.36556683936516710509513459177, −6.23916253747487226213303530638, −5.32359514577515486715068028075, −3.20446852748645676549388394871, 0.25852575978823346909573396353, 2.53012790942622937604874157457, 3.59951872493571199716439445806, 5.37991315035471063801366666066, 7.16760463541432259765514975770, 8.475256643206997600284867986081, 9.224340233120072038634157303472, 10.37465434400901984673741307918, 10.74780540115663247756279378579, 12.14330004486223150510423024135

Graph of the ZZ-function along the critical line