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Properties

Label 2-91-91.81-c1-0-4
Degree 22
Conductor 9191
Sign −0.803+0.595i-0.803 + 0.595i
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s − 3·3-s + (0.500 + 0.866i)4-s + (−1.5 − 2.59i)5-s + (1.5 − 2.59i)6-s + (−2.5 − 0.866i)7-s − 3·8-s + 6·9-s + 3·10-s − 3·11-s + (−1.50 − 2.59i)12-s + (−1 + 3.46i)13-s + (2 − 1.73i)14-s + (4.5 + 7.79i)15-s + (0.500 − 0.866i)16-s + (1 + 1.73i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s − 1.73·3-s + (0.250 + 0.433i)4-s + (−0.670 − 1.16i)5-s + (0.612 − 1.06i)6-s + (−0.944 − 0.327i)7-s − 1.06·8-s + 2·9-s + 0.948·10-s − 0.904·11-s + (−0.433 − 0.749i)12-s + (−0.277 + 0.960i)13-s + (0.534 − 0.462i)14-s + (1.16 + 2.01i)15-s + (0.125 − 0.216i)16-s + (0.242 + 0.420i)17-s + ⋯

Functional equation

Λ(s)=(91s/2 ΓC(s) L(s)=((−0.803+0.595i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(91s/2 ΓC(s+1/2) L(s)=((−0.803+0.595i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9191    =    7⋅137 \cdot 13
Sign: −0.803+0.595i-0.803 + 0.595i
Analytic conductor: 0.7266380.726638
Root analytic conductor: 0.8524310.852431
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ91(81,⋅)\chi_{91} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 91, ( :1/2), −0.803+0.595i)(2,\ 91,\ (\ :1/2),\ -0.803 + 0.595i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+(2.5+0.866i)T 1 + (2.5 + 0.866i)T
13 1+(1−3.46i)T 1 + (1 - 3.46i)T
good2 1+(0.5−0.866i)T+(−1−1.73i)T2 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2}
3 1+3T+3T2 1 + 3T + 3T^{2}
5 1+(1.5+2.59i)T+(−2.5+4.33i)T2 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2}
11 1+3T+11T2 1 + 3T + 11T^{2}
17 1+(−1−1.73i)T+(−8.5+14.7i)T2 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2}
19 1+T+19T2 1 + T + 19T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1+(3.5+6.06i)T+(−14.5+25.1i)T2 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2}
31 1+(1.5−2.59i)T+(−15.5−26.8i)T2 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2}
37 1+(1−1.73i)T+(−18.5−32.0i)T2 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.5+2.59i)T+(−20.5+35.5i)T2 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2}
43 1+(−3.5+6.06i)T+(−21.5−37.2i)T2 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2}
47 1+(0.5+0.866i)T+(−23.5+40.7i)T2 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.5−2.59i)T+(−26.5−45.8i)T2 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2}
59 1+(−2−3.46i)T+(−29.5+51.0i)T2 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2}
61 1+13T+61T2 1 + 13T + 61T^{2}
67 1+3T+67T2 1 + 3T + 67T^{2}
71 1+(6.5−11.2i)T+(−35.5−61.4i)T2 1 + (6.5 - 11.2i)T + (-35.5 - 61.4i)T^{2}
73 1+(−6.5+11.2i)T+(−36.5−63.2i)T2 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2}
79 1+(−1.5−2.59i)T+(−39.5+68.4i)T2 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(3−5.19i)T+(−44.5−77.0i)T2 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2}
97 1+(−2.5+4.33i)T+(−48.5−84.0i)T2 1 + (-2.5 + 4.33i)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

−13.17083072304890003935458042810, −12.35600926309536103741998797111, −11.82021533416337375398473062326, −10.54540591994848998103598344623, −9.180931829567947556896723749772, −7.74588984405759713913977021044, −6.70721235102570303894165474320, −5.59003972084021475716683567771, −4.19143906619076892162692347148, 0, 3.04189453050271718744975784464, 5.38223469062928345016302182981, 6.33600266034391586792357201625, 7.37774941531169181062342851270, 9.710119850407055439006879556857, 10.61928492568430797996677418294, 11.03561694622216597655332642963, 12.08027994337324273952628204747, 12.87148941030843382217326914293

Graph of the ZZ-function along the critical line