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Properties

Label 2-91-13.9-c1-0-5
Degree 22
Conductor 9191
Sign 0.822+0.568i0.822 + 0.568i
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  + (−0.115 + 0.200i)2-s + (1.66 − 2.87i)3-s + (0.973 + 1.68i)4-s − 2.23·5-s + (0.384 + 0.665i)6-s + (0.5 + 0.866i)7-s − 0.913·8-s + (−4.01 − 6.96i)9-s + (0.258 − 0.447i)10-s + (−1.66 + 2.87i)11-s + 6.46·12-s + (3.40 + 1.19i)13-s − 0.231·14-s + (−3.70 + 6.41i)15-s + (−1.84 + 3.18i)16-s + (0.687 + 1.19i)17-s + ⋯
L(s)  = 1  + (−0.0817 + 0.141i)2-s + (0.959 − 1.66i)3-s + (0.486 + 0.842i)4-s − 0.997·5-s + (0.156 + 0.271i)6-s + (0.188 + 0.327i)7-s − 0.322·8-s + (−1.33 − 2.32i)9-s + (0.0816 − 0.141i)10-s + (−0.500 + 0.867i)11-s + 1.86·12-s + (0.943 + 0.330i)13-s − 0.0618·14-s + (−0.957 + 1.65i)15-s + (−0.460 + 0.797i)16-s + (0.166 + 0.288i)17-s + ⋯

Functional equation

Λ(s)=(91s/2 ΓC(s) L(s)=((0.822+0.568i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(91s/2 ΓC(s+1/2) L(s)=((0.822+0.568i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9191    =    7⋅137 \cdot 13
Sign: 0.822+0.568i0.822 + 0.568i
Analytic conductor: 0.7266380.726638
Root analytic conductor: 0.8524310.852431
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ91(22,⋅)\chi_{91} (22, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 91, ( :1/2), 0.822+0.568i)(2,\ 91,\ (\ :1/2),\ 0.822 + 0.568i)

Particular Values

L(1)L(1) ≈\approx 1.09941−0.343098i1.09941 - 0.343098i
L(12)L(\frac12) ≈\approx 1.09941−0.343098i1.09941 - 0.343098i
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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
13 1+(−3.40−1.19i)T 1 + (-3.40 - 1.19i)T
good2 1+(0.115−0.200i)T+(−1−1.73i)T2 1 + (0.115 - 0.200i)T + (-1 - 1.73i)T^{2}
3 1+(−1.66+2.87i)T+(−1.5−2.59i)T2 1 + (-1.66 + 2.87i)T + (-1.5 - 2.59i)T^{2}
5 1+2.23T+5T2 1 + 2.23T + 5T^{2}
11 1+(1.66−2.87i)T+(−5.5−9.52i)T2 1 + (1.66 - 2.87i)T + (-5.5 - 9.52i)T^{2}
17 1+(−0.687−1.19i)T+(−8.5+14.7i)T2 1 + (-0.687 - 1.19i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.61+2.80i)T+(−9.5+16.4i)T2 1 + (1.61 + 2.80i)T + (-9.5 + 16.4i)T^{2}
23 1+(0.419−0.726i)T+(−11.5−19.9i)T2 1 + (0.419 - 0.726i)T + (-11.5 - 19.9i)T^{2}
29 1+(−0.303+0.525i)T+(−14.5−25.1i)T2 1 + (-0.303 + 0.525i)T + (-14.5 - 25.1i)T^{2}
31 1−1.71T+31T2 1 - 1.71T + 31T^{2}
37 1+(0.776−1.34i)T+(−18.5−32.0i)T2 1 + (0.776 - 1.34i)T + (-18.5 - 32.0i)T^{2}
41 1+(−4.58+7.94i)T+(−20.5−35.5i)T2 1 + (-4.58 + 7.94i)T + (-20.5 - 35.5i)T^{2}
43 1+(0.615+1.06i)T+(−21.5+37.2i)T2 1 + (0.615 + 1.06i)T + (-21.5 + 37.2i)T^{2}
47 1+1.62T+47T2 1 + 1.62T + 47T^{2}
53 1−8.39T+53T2 1 - 8.39T + 53T^{2}
59 1+(4.41+7.64i)T+(−29.5+51.0i)T2 1 + (4.41 + 7.64i)T + (-29.5 + 51.0i)T^{2}
61 1+(2.73+4.73i)T+(−30.5+52.8i)T2 1 + (2.73 + 4.73i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.09+8.82i)T+(−33.5−58.0i)T2 1 + (-5.09 + 8.82i)T + (-33.5 - 58.0i)T^{2}
71 1+(−2.60−4.51i)T+(−35.5+61.4i)T2 1 + (-2.60 - 4.51i)T + (-35.5 + 61.4i)T^{2}
73 1+3.96T+73T2 1 + 3.96T + 73T^{2}
79 1−6.45T+79T2 1 - 6.45T + 79T^{2}
83 1+4.64T+83T2 1 + 4.64T + 83T^{2}
89 1+(4.56−7.90i)T+(−44.5−77.0i)T2 1 + (4.56 - 7.90i)T + (-44.5 - 77.0i)T^{2}
97 1+(−7.67−13.3i)T+(−48.5+84.0i)T2 1 + (-7.67 - 13.3i)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

−13.78207974729377296298621016257, −12.78351063907466474921411146983, −12.17182714957777213462043583243, −11.29642291855801554575648955137, −8.915719136285082260029157961502, −8.095004957867821633147710496278, −7.43293287273454841364254687502, −6.46496255287246761462898355204, −3.67193402793460650513997232666, −2.23590199528254669182359196927, 3.06429129986526176513509224090, 4.25978974538480053534653495049, 5.68597288456527941638358916005, 7.86676651477109985167543999693, 8.741081848373144348222561625066, 10.03768831527980089678828627421, 10.78261554121269469031982695617, 11.48345058769687307890730478741, 13.58422793346959703032198834412, 14.51247349698249085364381465123

Graph of the ZZ-function along the critical line