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Properties

Label 2-59-59.58-c6-0-23
Degree 22
Conductor 5959
Sign −0.843+0.536i-0.843 + 0.536i
Analytic cond. 13.573113.5731
Root an. cond. 3.684183.68418
Motivic weight 66
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.3i·2-s + 12.0·3-s − 44.1·4-s − 27.1·5-s − 125. i·6-s + 631.·7-s − 206. i·8-s − 583.·9-s + 282. i·10-s − 1.42e3i·11-s − 532.·12-s − 1.31e3i·13-s − 6.56e3i·14-s − 327.·15-s − 4.97e3·16-s + 3.33e3·17-s + ⋯
L(s)  = 1  − 1.29i·2-s + 0.447·3-s − 0.689·4-s − 0.217·5-s − 0.581i·6-s + 1.84·7-s − 0.403i·8-s − 0.800·9-s + 0.282i·10-s − 1.06i·11-s − 0.308·12-s − 0.599i·13-s − 2.39i·14-s − 0.0970·15-s − 1.21·16-s + 0.678·17-s + ⋯

Functional equation

Λ(s)=(59s/2 ΓC(s) L(s)=((−0.843+0.536i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 + 0.536i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(59s/2 ΓC(s+3) L(s)=((−0.843+0.536i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.843 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5959
Sign: −0.843+0.536i-0.843 + 0.536i
Analytic conductor: 13.573113.5731
Root analytic conductor: 3.684183.68418
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ59(58,⋅)\chi_{59} (58, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 59, ( :3), −0.843+0.536i)(2,\ 59,\ (\ :3),\ -0.843 + 0.536i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.616357−2.11859i0.616357 - 2.11859i
L(12)L(\frac12) ≈\approx 0.616357−2.11859i0.616357 - 2.11859i
L(4)L(4) 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)
bad59 1+(−1.73e5+1.10e5i)T 1 + (-1.73e5 + 1.10e5i)T
good2 1+10.3iT−64T2 1 + 10.3iT - 64T^{2}
3 1−12.0T+729T2 1 - 12.0T + 729T^{2}
5 1+27.1T+1.56e4T2 1 + 27.1T + 1.56e4T^{2}
7 1−631.T+1.17e5T2 1 - 631.T + 1.17e5T^{2}
11 1+1.42e3iT−1.77e6T2 1 + 1.42e3iT - 1.77e6T^{2}
13 1+1.31e3iT−4.82e6T2 1 + 1.31e3iT - 4.82e6T^{2}
17 1−3.33e3T+2.41e7T2 1 - 3.33e3T + 2.41e7T^{2}
19 1−1.22e3T+4.70e7T2 1 - 1.22e3T + 4.70e7T^{2}
23 1−141.iT−1.48e8T2 1 - 141. iT - 1.48e8T^{2}
29 1−798.T+5.94e8T2 1 - 798.T + 5.94e8T^{2}
31 1+2.40e4iT−8.87e8T2 1 + 2.40e4iT - 8.87e8T^{2}
37 1−7.66e4iT−2.56e9T2 1 - 7.66e4iT - 2.56e9T^{2}
41 1−6.84e4T+4.75e9T2 1 - 6.84e4T + 4.75e9T^{2}
43 1+4.06e4iT−6.32e9T2 1 + 4.06e4iT - 6.32e9T^{2}
47 1+1.48e5iT−1.07e10T2 1 + 1.48e5iT - 1.07e10T^{2}
53 1−1.58e5T+2.21e10T2 1 - 1.58e5T + 2.21e10T^{2}
61 1−1.78e5iT−5.15e10T2 1 - 1.78e5iT - 5.15e10T^{2}
67 1−1.92e5iT−9.04e10T2 1 - 1.92e5iT - 9.04e10T^{2}
71 1+1.64e5T+1.28e11T2 1 + 1.64e5T + 1.28e11T^{2}
73 1−6.68e5iT−1.51e11T2 1 - 6.68e5iT - 1.51e11T^{2}
79 1−3.50e5T+2.43e11T2 1 - 3.50e5T + 2.43e11T^{2}
83 1−6.24e4iT−3.26e11T2 1 - 6.24e4iT - 3.26e11T^{2}
89 1−7.03e5iT−4.96e11T2 1 - 7.03e5iT - 4.96e11T^{2}
97 1+1.11e6iT−8.32e11T2 1 + 1.11e6iT - 8.32e11T^{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

−13.40602048254544278632598331444, −11.72198888491125519978992416164, −11.43999852210106497769370452721, −10.27384407455061716695338823799, −8.684897137069266513299297416097, −7.83987169415959283214196657921, −5.47501255585419372775794687249, −3.77590529451141383524395423269, −2.41096016798192428979098829323, −0.921044184957557780408251163895, 2.01651397261087380791126460585, 4.50560612409528512465216074277, 5.65494538453628551524196093032, 7.40505550735955109913183609263, 8.020541231594115899557051958997, 9.118723957007729734234145191227, 11.05729736128249683291228860009, 12.00613150354401551075373867847, 14.00633936708063670418203183272, 14.48730194243056004499344964236

Graph of the ZZ-function along the critical line