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Properties

Label 2-189-189.101-c1-0-12
Degree 22
Conductor 189189
Sign 0.619−0.785i0.619 - 0.785i
Analytic cond. 1.509171.50917
Root an. cond. 1.228481.22848
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  + (−1.08 + 1.29i)2-s + (1.72 − 0.201i)3-s + (−0.151 − 0.861i)4-s + (1.13 − 0.954i)5-s + (−1.61 + 2.45i)6-s + (2.51 − 0.806i)7-s + (−1.65 − 0.953i)8-s + (2.91 − 0.693i)9-s + 2.51i·10-s + (−0.0836 + 0.0996i)11-s + (−0.435 − 1.45i)12-s + (−0.311 + 0.855i)13-s + (−1.69 + 4.15i)14-s + (1.76 − 1.87i)15-s + (4.68 − 1.70i)16-s − 5.63·17-s + ⋯
L(s)  = 1  + (−0.770 + 0.918i)2-s + (0.993 − 0.116i)3-s + (−0.0759 − 0.430i)4-s + (0.508 − 0.426i)5-s + (−0.658 + 1.00i)6-s + (0.952 − 0.304i)7-s + (−0.584 − 0.337i)8-s + (0.972 − 0.231i)9-s + 0.796i·10-s + (−0.0252 + 0.0300i)11-s + (−0.125 − 0.419i)12-s + (−0.0863 + 0.237i)13-s + (−0.453 + 1.10i)14-s + (0.455 − 0.483i)15-s + (1.17 − 0.426i)16-s − 1.36·17-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓC(s) L(s)=((0.619−0.785i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(189s/2 ΓC(s+1/2) L(s)=((0.619−0.785i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: 0.619−0.785i0.619 - 0.785i
Analytic conductor: 1.509171.50917
Root analytic conductor: 1.228481.22848
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ189(101,⋅)\chi_{189} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 189, ( :1/2), 0.619−0.785i)(2,\ 189,\ (\ :1/2),\ 0.619 - 0.785i)

Particular Values

L(1)L(1) ≈\approx 1.09431+0.530510i1.09431 + 0.530510i
L(12)L(\frac12) ≈\approx 1.09431+0.530510i1.09431 + 0.530510i
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+(−1.72+0.201i)T 1 + (-1.72 + 0.201i)T
7 1+(−2.51+0.806i)T 1 + (-2.51 + 0.806i)T
good2 1+(1.08−1.29i)T+(−0.347−1.96i)T2 1 + (1.08 - 1.29i)T + (-0.347 - 1.96i)T^{2}
5 1+(−1.13+0.954i)T+(0.868−4.92i)T2 1 + (-1.13 + 0.954i)T + (0.868 - 4.92i)T^{2}
11 1+(0.0836−0.0996i)T+(−1.91−10.8i)T2 1 + (0.0836 - 0.0996i)T + (-1.91 - 10.8i)T^{2}
13 1+(0.311−0.855i)T+(−9.95−8.35i)T2 1 + (0.311 - 0.855i)T + (-9.95 - 8.35i)T^{2}
17 1+5.63T+17T2 1 + 5.63T + 17T^{2}
19 1−0.0959iT−19T2 1 - 0.0959iT - 19T^{2}
23 1+(2.22−6.10i)T+(−17.6−14.7i)T2 1 + (2.22 - 6.10i)T + (-17.6 - 14.7i)T^{2}
29 1+(−0.238−0.656i)T+(−22.2+18.6i)T2 1 + (-0.238 - 0.656i)T + (-22.2 + 18.6i)T^{2}
31 1+(8.96−1.58i)T+(29.1−10.6i)T2 1 + (8.96 - 1.58i)T + (29.1 - 10.6i)T^{2}
37 1+(−1.72+2.98i)T+(−18.5−32.0i)T2 1 + (-1.72 + 2.98i)T + (-18.5 - 32.0i)T^{2}
41 1+(2.04+0.744i)T+(31.4+26.3i)T2 1 + (2.04 + 0.744i)T + (31.4 + 26.3i)T^{2}
43 1+(−1.37+7.82i)T+(−40.4−14.7i)T2 1 + (-1.37 + 7.82i)T + (-40.4 - 14.7i)T^{2}
47 1+(−2.18+12.3i)T+(−44.1−16.0i)T2 1 + (-2.18 + 12.3i)T + (-44.1 - 16.0i)T^{2}
53 1+(−1.26−0.731i)T+(26.5+45.8i)T2 1 + (-1.26 - 0.731i)T + (26.5 + 45.8i)T^{2}
59 1+(−5.96−2.17i)T+(45.1+37.9i)T2 1 + (-5.96 - 2.17i)T + (45.1 + 37.9i)T^{2}
61 1+(7.69+1.35i)T+(57.3+20.8i)T2 1 + (7.69 + 1.35i)T + (57.3 + 20.8i)T^{2}
67 1+(11.0−9.25i)T+(11.6−65.9i)T2 1 + (11.0 - 9.25i)T + (11.6 - 65.9i)T^{2}
71 1+(−2.38+1.37i)T+(35.5−61.4i)T2 1 + (-2.38 + 1.37i)T + (35.5 - 61.4i)T^{2}
73 1+(6.94−4.01i)T+(36.5−63.2i)T2 1 + (6.94 - 4.01i)T + (36.5 - 63.2i)T^{2}
79 1+(−0.310−0.260i)T+(13.7+77.7i)T2 1 + (-0.310 - 0.260i)T + (13.7 + 77.7i)T^{2}
83 1+(4.54−1.65i)T+(63.5−53.3i)T2 1 + (4.54 - 1.65i)T + (63.5 - 53.3i)T^{2}
89 1−15.9T+89T2 1 - 15.9T + 89T^{2}
97 1+(−10.2−1.81i)T+(91.1+33.1i)T2 1 + (-10.2 - 1.81i)T + (91.1 + 33.1i)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

−13.03518090119010279427711783602, −11.74861207096662387168573581941, −10.34840235388617404426882658093, −9.122713574554455470153452669219, −8.795787763496253921779273014659, −7.63392040422198378088405049055, −7.00332979829701178425121906783, −5.44839869583452874166247329659, −3.87642844258091703383201254136, −1.86842985824864306189332680960, 1.88721534736917274968966389007, 2.74113380838063970774328684606, 4.51891069216117168150500758128, 6.21705635466459724088164977134, 7.79370671173691733725827452281, 8.701381926054362495900317930788, 9.424045402070332715215835840994, 10.48187076118919101940266323862, 11.05609990014616616183754328156, 12.27442853626385159834675061956

Graph of the ZZ-function along the critical line