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Properties

Label 2-189-189.101-c1-0-18
Degree 22
Conductor 189189
Sign −0.298+0.954i-0.298 + 0.954i
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  + (1.75 − 2.08i)2-s + (−1.10 + 1.33i)3-s + (−0.940 − 5.33i)4-s + (2.08 − 1.74i)5-s + (0.856 + 4.63i)6-s + (−0.122 + 2.64i)7-s + (−8.05 − 4.65i)8-s + (−0.568 − 2.94i)9-s − 7.40i·10-s + (0.0911 − 0.108i)11-s + (8.16 + 4.62i)12-s + (−0.695 + 1.91i)13-s + (5.29 + 4.88i)14-s + (0.0376 + 4.71i)15-s + (−13.6 + 4.96i)16-s + 4.00·17-s + ⋯
L(s)  = 1  + (1.23 − 1.47i)2-s + (−0.636 + 0.771i)3-s + (−0.470 − 2.66i)4-s + (0.931 − 0.781i)5-s + (0.349 + 1.89i)6-s + (−0.0463 + 0.998i)7-s + (−2.84 − 1.64i)8-s + (−0.189 − 0.981i)9-s − 2.34i·10-s + (0.0274 − 0.0327i)11-s + (2.35 + 1.33i)12-s + (−0.192 + 0.530i)13-s + (1.41 + 1.30i)14-s + (0.00971 + 1.21i)15-s + (−3.40 + 1.24i)16-s + 0.970·17-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓC(s) L(s)=((−0.298+0.954i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(189s/2 ΓC(s+1/2) L(s)=((−0.298+0.954i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: −0.298+0.954i-0.298 + 0.954i
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), −0.298+0.954i)(2,\ 189,\ (\ :1/2),\ -0.298 + 0.954i)

Particular Values

L(1)L(1) ≈\approx 1.10864−1.50800i1.10864 - 1.50800i
L(12)L(\frac12) ≈\approx 1.10864−1.50800i1.10864 - 1.50800i
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+(1.10−1.33i)T 1 + (1.10 - 1.33i)T
7 1+(0.122−2.64i)T 1 + (0.122 - 2.64i)T
good2 1+(−1.75+2.08i)T+(−0.347−1.96i)T2 1 + (-1.75 + 2.08i)T + (-0.347 - 1.96i)T^{2}
5 1+(−2.08+1.74i)T+(0.868−4.92i)T2 1 + (-2.08 + 1.74i)T + (0.868 - 4.92i)T^{2}
11 1+(−0.0911+0.108i)T+(−1.91−10.8i)T2 1 + (-0.0911 + 0.108i)T + (-1.91 - 10.8i)T^{2}
13 1+(0.695−1.91i)T+(−9.95−8.35i)T2 1 + (0.695 - 1.91i)T + (-9.95 - 8.35i)T^{2}
17 1−4.00T+17T2 1 - 4.00T + 17T^{2}
19 1−3.04iT−19T2 1 - 3.04iT - 19T^{2}
23 1+(0.449−1.23i)T+(−17.6−14.7i)T2 1 + (0.449 - 1.23i)T + (-17.6 - 14.7i)T^{2}
29 1+(−2.12−5.84i)T+(−22.2+18.6i)T2 1 + (-2.12 - 5.84i)T + (-22.2 + 18.6i)T^{2}
31 1+(4.18−0.738i)T+(29.1−10.6i)T2 1 + (4.18 - 0.738i)T + (29.1 - 10.6i)T^{2}
37 1+(−4.69+8.12i)T+(−18.5−32.0i)T2 1 + (-4.69 + 8.12i)T + (-18.5 - 32.0i)T^{2}
41 1+(−0.303−0.110i)T+(31.4+26.3i)T2 1 + (-0.303 - 0.110i)T + (31.4 + 26.3i)T^{2}
43 1+(−0.643+3.65i)T+(−40.4−14.7i)T2 1 + (-0.643 + 3.65i)T + (-40.4 - 14.7i)T^{2}
47 1+(1.15−6.57i)T+(−44.1−16.0i)T2 1 + (1.15 - 6.57i)T + (-44.1 - 16.0i)T^{2}
53 1+(6.59+3.81i)T+(26.5+45.8i)T2 1 + (6.59 + 3.81i)T + (26.5 + 45.8i)T^{2}
59 1+(6.06+2.20i)T+(45.1+37.9i)T2 1 + (6.06 + 2.20i)T + (45.1 + 37.9i)T^{2}
61 1+(11.1+1.96i)T+(57.3+20.8i)T2 1 + (11.1 + 1.96i)T + (57.3 + 20.8i)T^{2}
67 1+(1.37−1.15i)T+(11.6−65.9i)T2 1 + (1.37 - 1.15i)T + (11.6 - 65.9i)T^{2}
71 1+(−6.03+3.48i)T+(35.5−61.4i)T2 1 + (-6.03 + 3.48i)T + (35.5 - 61.4i)T^{2}
73 1+(−3.67+2.12i)T+(36.5−63.2i)T2 1 + (-3.67 + 2.12i)T + (36.5 - 63.2i)T^{2}
79 1+(11.4+9.62i)T+(13.7+77.7i)T2 1 + (11.4 + 9.62i)T + (13.7 + 77.7i)T^{2}
83 1+(0.966−0.351i)T+(63.5−53.3i)T2 1 + (0.966 - 0.351i)T + (63.5 - 53.3i)T^{2}
89 1+5.17T+89T2 1 + 5.17T + 89T^{2}
97 1+(7.25+1.27i)T+(91.1+33.1i)T2 1 + (7.25 + 1.27i)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

−12.42043203818800932923441099670, −11.48895468415244982903365169901, −10.53096926857948685108813099883, −9.544980974801195547255002714015, −9.133907656486211128037052225426, −6.06456160636455591706472921716, −5.50725270754809051139528324921, −4.67594730656399651065583312204, −3.29857073796211077914382443059, −1.67082978424396752150744859438, 2.92935547350536007070193442701, 4.61504876128693274895956745291, 5.77877425320820477227379326674, 6.47504076346463991882204836245, 7.29971313239781833650008855071, 8.066554144042852172113110129943, 9.957650225950072635861691293310, 11.21244036401565638948378504787, 12.38002695977875560154535424468, 13.24490384929582558676693654141

Graph of the ZZ-function along the critical line