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Properties

Label 2-92-23.18-c5-0-7
Degree 22
Conductor 9292
Sign −0.583+0.812i-0.583 + 0.812i
Analytic cond. 14.755314.7553
Root an. cond. 3.841263.84126
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.49 + 4.03i)3-s + (−12.4 + 86.8i)5-s + (−77.3 − 169. i)7-s + (30.5 + 212. i)9-s + (−341. + 100. i)11-s + (260. − 569. i)13-s + (−306. − 354. i)15-s + (1.66e3 − 1.06e3i)17-s + (−1.79e3 − 1.15e3i)19-s + (953. + 280. i)21-s + (−2.48e3 + 532. i)23-s + (−4.39e3 − 1.29e3i)25-s + (−2.05e3 − 1.31e3i)27-s + (−1.14e3 + 733. i)29-s + (−4.54e3 − 5.25e3i)31-s + ⋯
L(s)  = 1  + (−0.224 + 0.258i)3-s + (−0.223 + 1.55i)5-s + (−0.596 − 1.30i)7-s + (0.125 + 0.873i)9-s + (−0.851 + 0.249i)11-s + (0.427 − 0.935i)13-s + (−0.352 − 0.406i)15-s + (1.39 − 0.897i)17-s + (−1.14 − 0.734i)19-s + (0.471 + 0.138i)21-s + (−0.977 + 0.209i)23-s + (−1.40 − 0.413i)25-s + (−0.542 − 0.348i)27-s + (−0.251 + 0.161i)29-s + (−0.850 − 0.981i)31-s + ⋯

Functional equation

Λ(s)=(92s/2 ΓC(s) L(s)=((−0.583+0.812i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 + 0.812i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(92s/2 ΓC(s+5/2) L(s)=((−0.583+0.812i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9292    =    22⋅232^{2} \cdot 23
Sign: −0.583+0.812i-0.583 + 0.812i
Analytic conductor: 14.755314.7553
Root analytic conductor: 3.841263.84126
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ92(41,⋅)\chi_{92} (41, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 92, ( :5/2), −0.583+0.812i)(2,\ 92,\ (\ :5/2),\ -0.583 + 0.812i)

Particular Values

L(3)L(3) ≈\approx 0.0980892−0.191116i0.0980892 - 0.191116i
L(12)L(\frac12) ≈\approx 0.0980892−0.191116i0.0980892 - 0.191116i
L(72)L(\frac{7}{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)
bad2 1 1
23 1+(2.48e3−532.i)T 1 + (2.48e3 - 532. i)T
good3 1+(3.49−4.03i)T+(−34.5−240.i)T2 1 + (3.49 - 4.03i)T + (-34.5 - 240. i)T^{2}
5 1+(12.4−86.8i)T+(−2.99e3−880.i)T2 1 + (12.4 - 86.8i)T + (-2.99e3 - 880. i)T^{2}
7 1+(77.3+169.i)T+(−1.10e4+1.27e4i)T2 1 + (77.3 + 169. i)T + (-1.10e4 + 1.27e4i)T^{2}
11 1+(341.−100.i)T+(1.35e5−8.70e4i)T2 1 + (341. - 100. i)T + (1.35e5 - 8.70e4i)T^{2}
13 1+(−260.+569.i)T+(−2.43e5−2.80e5i)T2 1 + (-260. + 569. i)T + (-2.43e5 - 2.80e5i)T^{2}
17 1+(−1.66e3+1.06e3i)T+(5.89e5−1.29e6i)T2 1 + (-1.66e3 + 1.06e3i)T + (5.89e5 - 1.29e6i)T^{2}
19 1+(1.79e3+1.15e3i)T+(1.02e6+2.25e6i)T2 1 + (1.79e3 + 1.15e3i)T + (1.02e6 + 2.25e6i)T^{2}
29 1+(1.14e3−733.i)T+(8.52e6−1.86e7i)T2 1 + (1.14e3 - 733. i)T + (8.52e6 - 1.86e7i)T^{2}
31 1+(4.54e3+5.25e3i)T+(−4.07e6+2.83e7i)T2 1 + (4.54e3 + 5.25e3i)T + (-4.07e6 + 2.83e7i)T^{2}
37 1+(1.69e3+1.18e4i)T+(−6.65e7+1.95e7i)T2 1 + (1.69e3 + 1.18e4i)T + (-6.65e7 + 1.95e7i)T^{2}
41 1+(1.36e3−9.47e3i)T+(−1.11e8−3.26e7i)T2 1 + (1.36e3 - 9.47e3i)T + (-1.11e8 - 3.26e7i)T^{2}
43 1+(−1.78e3+2.05e3i)T+(−2.09e7−1.45e8i)T2 1 + (-1.78e3 + 2.05e3i)T + (-2.09e7 - 1.45e8i)T^{2}
47 1+2.52e3T+2.29e8T2 1 + 2.52e3T + 2.29e8T^{2}
53 1+(−7.59e3−1.66e4i)T+(−2.73e8+3.16e8i)T2 1 + (-7.59e3 - 1.66e4i)T + (-2.73e8 + 3.16e8i)T^{2}
59 1+(8.39e3−1.83e4i)T+(−4.68e8−5.40e8i)T2 1 + (8.39e3 - 1.83e4i)T + (-4.68e8 - 5.40e8i)T^{2}
61 1+(−2.02e4−2.33e4i)T+(−1.20e8+8.35e8i)T2 1 + (-2.02e4 - 2.33e4i)T + (-1.20e8 + 8.35e8i)T^{2}
67 1+(−6.47e4−1.90e4i)T+(1.13e9+7.29e8i)T2 1 + (-6.47e4 - 1.90e4i)T + (1.13e9 + 7.29e8i)T^{2}
71 1+(5.71e4+1.67e4i)T+(1.51e9+9.75e8i)T2 1 + (5.71e4 + 1.67e4i)T + (1.51e9 + 9.75e8i)T^{2}
73 1+(6.82e4+4.38e4i)T+(8.61e8+1.88e9i)T2 1 + (6.82e4 + 4.38e4i)T + (8.61e8 + 1.88e9i)T^{2}
79 1+(3.39e4−7.44e4i)T+(−2.01e9−2.32e9i)T2 1 + (3.39e4 - 7.44e4i)T + (-2.01e9 - 2.32e9i)T^{2}
83 1+(1.34e3+9.32e3i)T+(−3.77e9+1.10e9i)T2 1 + (1.34e3 + 9.32e3i)T + (-3.77e9 + 1.10e9i)T^{2}
89 1+(2.45e3−2.83e3i)T+(−7.94e8−5.52e9i)T2 1 + (2.45e3 - 2.83e3i)T + (-7.94e8 - 5.52e9i)T^{2}
97 1+(586.−4.08e3i)T+(−8.23e9−2.41e9i)T2 1 + (586. - 4.08e3i)T + (-8.23e9 - 2.41e9i)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

−12.95347032972236624181934641136, −11.22434891676474053598539702895, −10.48878644164788898445466335521, −10.02738933667835830825465142751, −7.72964380813299414117373820351, −7.21559759198490631040393957364, −5.71653053824520360171200634251, −3.96239014093520708439183137389, −2.71093686263478497429868535632, −0.086885101840385521981645747357, 1.62884879599496011199679969607, 3.76456460125041633949977201117, 5.40305442811258417981378240387, 6.26068756519420728249382706414, 8.228239483895275777231970297586, 8.872936633517850915043977565247, 10.00758252263983240412575883211, 11.82505688322042916678818431102, 12.46296663636994886723186180127, 12.95848685928459114526641019960

Graph of the ZZ-function along the critical line