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Properties

Label 2-91-13.10-c1-0-0
Degree 22
Conductor 9191
Sign −0.383−0.923i-0.383 - 0.923i
Analytic cond. 0.7266380.726638
Root an. cond. 0.8524310.852431
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  + (−1.20 + 0.692i)2-s + (1.41 + 2.44i)3-s + (−0.0395 + 0.0685i)4-s − 0.518i·5-s + (−3.39 − 1.95i)6-s + (−0.866 − 0.5i)7-s − 2.88i·8-s + (−2.49 + 4.31i)9-s + (0.359 + 0.622i)10-s + (1.40 − 0.812i)11-s − 0.223·12-s + (1.42 + 3.31i)13-s + 1.38·14-s + (1.26 − 0.733i)15-s + (1.91 + 3.32i)16-s + (0.974 − 1.68i)17-s + ⋯
L(s)  = 1  + (−0.848 + 0.490i)2-s + (0.815 + 1.41i)3-s + (−0.0197 + 0.0342i)4-s − 0.232i·5-s + (−1.38 − 0.799i)6-s + (−0.327 − 0.188i)7-s − 1.01i·8-s + (−0.830 + 1.43i)9-s + (0.113 + 0.196i)10-s + (0.424 − 0.244i)11-s − 0.0645·12-s + (0.395 + 0.918i)13-s + 0.370·14-s + (0.327 − 0.189i)15-s + (0.479 + 0.830i)16-s + (0.236 − 0.409i)17-s + ⋯

Functional equation

Λ(s)=(91s/2 ΓC(s) L(s)=((−0.383−0.923i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(91s/2 ΓC(s+1/2) L(s)=((−0.383−0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9191    =    7⋅137 \cdot 13
Sign: −0.383−0.923i-0.383 - 0.923i
Analytic conductor: 0.7266380.726638
Root analytic conductor: 0.8524310.852431
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ91(36,⋅)\chi_{91} (36, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 91, ( :1/2), −0.383−0.923i)(2,\ 91,\ (\ :1/2),\ -0.383 - 0.923i)

Particular Values

L(1)L(1) ≈\approx 0.432642+0.648347i0.432642 + 0.648347i
L(12)L(\frac12) ≈\approx 0.432642+0.648347i0.432642 + 0.648347i
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)
bad7 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
13 1+(−1.42−3.31i)T 1 + (-1.42 - 3.31i)T
good2 1+(1.20−0.692i)T+(1−1.73i)T2 1 + (1.20 - 0.692i)T + (1 - 1.73i)T^{2}
3 1+(−1.41−2.44i)T+(−1.5+2.59i)T2 1 + (-1.41 - 2.44i)T + (-1.5 + 2.59i)T^{2}
5 1+0.518iT−5T2 1 + 0.518iT - 5T^{2}
11 1+(−1.40+0.812i)T+(5.5−9.52i)T2 1 + (-1.40 + 0.812i)T + (5.5 - 9.52i)T^{2}
17 1+(−0.974+1.68i)T+(−8.5−14.7i)T2 1 + (-0.974 + 1.68i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.15−1.24i)T+(9.5+16.4i)T2 1 + (-2.15 - 1.24i)T + (9.5 + 16.4i)T^{2}
23 1+(4.57+7.91i)T+(−11.5+19.9i)T2 1 + (4.57 + 7.91i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.61−4.52i)T+(−14.5+25.1i)T2 1 + (-2.61 - 4.52i)T + (-14.5 + 25.1i)T^{2}
31 1+5.79iT−31T2 1 + 5.79iT - 31T^{2}
37 1+(8.85−5.11i)T+(18.5−32.0i)T2 1 + (8.85 - 5.11i)T + (18.5 - 32.0i)T^{2}
41 1+(−3.64+2.10i)T+(20.5−35.5i)T2 1 + (-3.64 + 2.10i)T + (20.5 - 35.5i)T^{2}
43 1+(0.498−0.863i)T+(−21.5−37.2i)T2 1 + (0.498 - 0.863i)T + (-21.5 - 37.2i)T^{2}
47 1+4.51iT−47T2 1 + 4.51iT - 47T^{2}
53 1+8.89T+53T2 1 + 8.89T + 53T^{2}
59 1+(5.37+3.10i)T+(29.5+51.0i)T2 1 + (5.37 + 3.10i)T + (29.5 + 51.0i)T^{2}
61 1+(−6.73+11.6i)T+(−30.5−52.8i)T2 1 + (-6.73 + 11.6i)T + (-30.5 - 52.8i)T^{2}
67 1+(7.25−4.18i)T+(33.5−58.0i)T2 1 + (7.25 - 4.18i)T + (33.5 - 58.0i)T^{2}
71 1+(4.50+2.59i)T+(35.5+61.4i)T2 1 + (4.50 + 2.59i)T + (35.5 + 61.4i)T^{2}
73 1+11.8iT−73T2 1 + 11.8iT - 73T^{2}
79 1−0.982T+79T2 1 - 0.982T + 79T^{2}
83 1−8.91iT−83T2 1 - 8.91iT - 83T^{2}
89 1+(10.4−6.00i)T+(44.5−77.0i)T2 1 + (10.4 - 6.00i)T + (44.5 - 77.0i)T^{2}
97 1+(−3.82−2.21i)T+(48.5+84.0i)T2 1 + (-3.82 - 2.21i)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

−14.56318564934029895581166298795, −13.77417865246612545996832831438, −12.27393795288480786418426784612, −10.64757254357678628578872970457, −9.701002453378307126912968250179, −8.962296011531587274047733800958, −8.203943044969050086982157961538, −6.63589867357642471318704246595, −4.54816144972298883613147924171, −3.43145108164336054455557537863, 1.45390583995939883720046521229, 3.02083321004080896155170308069, 5.83677852176420868914421980283, 7.28173288126409391739064551400, 8.260960236549061719177728715997, 9.188479679466817661547777377150, 10.34315023876456713962016632621, 11.69950957571609746995082346777, 12.67448349900212106314621502601, 13.75452316152219879934118989729

Graph of the ZZ-function along the critical line