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Properties

Label 2-91-13.9-c7-0-39
Degree 22
Conductor 9191
Sign −0.645+0.763i-0.645 + 0.763i
Analytic cond. 28.427028.4270
Root an. cond. 5.331705.33170
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.46 − 14.6i)2-s + (17.8 − 30.9i)3-s + (−79.4 − 137. i)4-s + 457.·5-s + (−302. − 524. i)6-s + (171.5 + 297. i)7-s − 522.·8-s + (454. + 786. i)9-s + (3.87e3 − 6.71e3i)10-s + (559. − 969. i)11-s − 5.68e3·12-s + (3.32e3 − 7.18e3i)13-s + 5.80e3·14-s + (8.17e3 − 1.41e4i)15-s + (5.73e3 − 9.94e3i)16-s + (4.75e3 + 8.24e3i)17-s + ⋯
L(s)  = 1  + (0.748 − 1.29i)2-s + (0.382 − 0.662i)3-s + (−0.620 − 1.07i)4-s + 1.63·5-s + (−0.572 − 0.991i)6-s + (0.188 + 0.327i)7-s − 0.361·8-s + (0.207 + 0.359i)9-s + (1.22 − 2.12i)10-s + (0.126 − 0.219i)11-s − 0.948·12-s + (0.420 − 0.907i)13-s + 0.565·14-s + (0.625 − 1.08i)15-s + (0.350 − 0.606i)16-s + (0.234 + 0.406i)17-s + ⋯

Functional equation

Λ(s)=(91s/2 ΓC(s) L(s)=((−0.645+0.763i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(91s/2 ΓC(s+7/2) L(s)=((−0.645+0.763i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9191    =    7⋅137 \cdot 13
Sign: −0.645+0.763i-0.645 + 0.763i
Analytic conductor: 28.427028.4270
Root analytic conductor: 5.331705.33170
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ91(22,⋅)\chi_{91} (22, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 91, ( :7/2), −0.645+0.763i)(2,\ 91,\ (\ :7/2),\ -0.645 + 0.763i)

Particular Values

L(4)L(4) ≈\approx 2.02713−4.36808i2.02713 - 4.36808i
L(12)L(\frac12) ≈\approx 2.02713−4.36808i2.02713 - 4.36808i
L(92)L(\frac{9}{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+(−171.5−297.i)T 1 + (-171.5 - 297. i)T
13 1+(−3.32e3+7.18e3i)T 1 + (-3.32e3 + 7.18e3i)T
good2 1+(−8.46+14.6i)T+(−64−110.i)T2 1 + (-8.46 + 14.6i)T + (-64 - 110. i)T^{2}
3 1+(−17.8+30.9i)T+(−1.09e3−1.89e3i)T2 1 + (-17.8 + 30.9i)T + (-1.09e3 - 1.89e3i)T^{2}
5 1−457.T+7.81e4T2 1 - 457.T + 7.81e4T^{2}
11 1+(−559.+969.i)T+(−9.74e6−1.68e7i)T2 1 + (-559. + 969. i)T + (-9.74e6 - 1.68e7i)T^{2}
17 1+(−4.75e3−8.24e3i)T+(−2.05e8+3.55e8i)T2 1 + (-4.75e3 - 8.24e3i)T + (-2.05e8 + 3.55e8i)T^{2}
19 1+(8.06e3+1.39e4i)T+(−4.46e8+7.74e8i)T2 1 + (8.06e3 + 1.39e4i)T + (-4.46e8 + 7.74e8i)T^{2}
23 1+(4.38e4−7.60e4i)T+(−1.70e9−2.94e9i)T2 1 + (4.38e4 - 7.60e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+(−5.58e3+9.67e3i)T+(−8.62e9−1.49e10i)T2 1 + (-5.58e3 + 9.67e3i)T + (-8.62e9 - 1.49e10i)T^{2}
31 1+2.91e5T+2.75e10T2 1 + 2.91e5T + 2.75e10T^{2}
37 1+(−9.17e4+1.58e5i)T+(−4.74e10−8.22e10i)T2 1 + (-9.17e4 + 1.58e5i)T + (-4.74e10 - 8.22e10i)T^{2}
41 1+(3.21e5−5.57e5i)T+(−9.73e10−1.68e11i)T2 1 + (3.21e5 - 5.57e5i)T + (-9.73e10 - 1.68e11i)T^{2}
43 1+(−2.72e5−4.71e5i)T+(−1.35e11+2.35e11i)T2 1 + (-2.72e5 - 4.71e5i)T + (-1.35e11 + 2.35e11i)T^{2}
47 1+1.22e6T+5.06e11T2 1 + 1.22e6T + 5.06e11T^{2}
53 1+8.49e5T+1.17e12T2 1 + 8.49e5T + 1.17e12T^{2}
59 1+(−1.50e6−2.60e6i)T+(−1.24e12+2.15e12i)T2 1 + (-1.50e6 - 2.60e6i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(1.76e6+3.06e6i)T+(−1.57e12+2.72e12i)T2 1 + (1.76e6 + 3.06e6i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(3.17e5−5.50e5i)T+(−3.03e12−5.24e12i)T2 1 + (3.17e5 - 5.50e5i)T + (-3.03e12 - 5.24e12i)T^{2}
71 1+(1.64e6+2.85e6i)T+(−4.54e12+7.87e12i)T2 1 + (1.64e6 + 2.85e6i)T + (-4.54e12 + 7.87e12i)T^{2}
73 1−3.29e6T+1.10e13T2 1 - 3.29e6T + 1.10e13T^{2}
79 1+5.67e5T+1.92e13T2 1 + 5.67e5T + 1.92e13T^{2}
83 1−5.08e6T+2.71e13T2 1 - 5.08e6T + 2.71e13T^{2}
89 1+(−2.71e6+4.70e6i)T+(−2.21e13−3.83e13i)T2 1 + (-2.71e6 + 4.70e6i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1+(−2.60e6−4.50e6i)T+(−4.03e13+6.99e13i)T2 1 + (-2.60e6 - 4.50e6i)T + (-4.03e13 + 6.99e13i)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

−12.72244775609821630214284228493, −11.25962818099701690021044024518, −10.32535183988840881856064668089, −9.363264625716165483305520082577, −7.83739071988535454920720865144, −6.07393890500774785260655504184, −5.06682653728595495946901885531, −3.21077810133642138776277039197, −2.02937939334443700673485214692, −1.39660366364747744871421019126, 1.79699264273918409910249780886, 3.83554752531985777535597243337, 4.96850631024781740130857911349, 6.13516251691608848712738347965, 6.94132154394453577796686293599, 8.608963066392313768323838534710, 9.626103139461429567512672405202, 10.56735694321476439203875885205, 12.52238745693670661578059803337, 13.58656658893372669120362747199

Graph of the ZZ-function along the critical line