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Properties

Label 2-92-92.19-c1-0-4
Degree 22
Conductor 9292
Sign 0.623−0.782i0.623 - 0.782i
Analytic cond. 0.7346230.734623
Root an. cond. 0.8571010.857101
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  + (1.14 + 0.830i)2-s + (−0.0248 − 0.0847i)3-s + (0.621 + 1.90i)4-s + (−1.32 + 2.06i)5-s + (0.0418 − 0.117i)6-s + (−0.647 − 4.50i)7-s + (−0.865 + 2.69i)8-s + (2.51 − 1.61i)9-s + (−3.23 + 1.26i)10-s + (−0.477 + 1.04i)11-s + (0.145 − 0.0999i)12-s + (0.686 − 4.77i)13-s + (2.99 − 5.69i)14-s + (0.208 + 0.0611i)15-s + (−3.22 + 2.36i)16-s + (−2.34 + 2.03i)17-s + ⋯
L(s)  = 1  + (0.809 + 0.586i)2-s + (−0.0143 − 0.0489i)3-s + (0.310 + 0.950i)4-s + (−0.594 + 0.924i)5-s + (0.0170 − 0.0480i)6-s + (−0.244 − 1.70i)7-s + (−0.306 + 0.951i)8-s + (0.839 − 0.539i)9-s + (−1.02 + 0.399i)10-s + (−0.144 + 0.315i)11-s + (0.0420 − 0.0288i)12-s + (0.190 − 1.32i)13-s + (0.801 − 1.52i)14-s + (0.0537 + 0.0157i)15-s + (−0.806 + 0.591i)16-s + (−0.569 + 0.493i)17-s + ⋯

Functional equation

Λ(s)=(92s/2 ΓC(s) L(s)=((0.623−0.782i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(92s/2 ΓC(s+1/2) L(s)=((0.623−0.782i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9292    =    22⋅232^{2} \cdot 23
Sign: 0.623−0.782i0.623 - 0.782i
Analytic conductor: 0.7346230.734623
Root analytic conductor: 0.8571010.857101
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ92(19,⋅)\chi_{92} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 92, ( :1/2), 0.623−0.782i)(2,\ 92,\ (\ :1/2),\ 0.623 - 0.782i)

Particular Values

L(1)L(1) ≈\approx 1.21155+0.583918i1.21155 + 0.583918i
L(12)L(\frac12) ≈\approx 1.21155+0.583918i1.21155 + 0.583918i
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)
bad2 1+(−1.14−0.830i)T 1 + (-1.14 - 0.830i)T
23 1+(−2.00+4.35i)T 1 + (-2.00 + 4.35i)T
good3 1+(0.0248+0.0847i)T+(−2.52+1.62i)T2 1 + (0.0248 + 0.0847i)T + (-2.52 + 1.62i)T^{2}
5 1+(1.32−2.06i)T+(−2.07−4.54i)T2 1 + (1.32 - 2.06i)T + (-2.07 - 4.54i)T^{2}
7 1+(0.647+4.50i)T+(−6.71+1.97i)T2 1 + (0.647 + 4.50i)T + (-6.71 + 1.97i)T^{2}
11 1+(0.477−1.04i)T+(−7.20−8.31i)T2 1 + (0.477 - 1.04i)T + (-7.20 - 8.31i)T^{2}
13 1+(−0.686+4.77i)T+(−12.4−3.66i)T2 1 + (-0.686 + 4.77i)T + (-12.4 - 3.66i)T^{2}
17 1+(2.34−2.03i)T+(2.41−16.8i)T2 1 + (2.34 - 2.03i)T + (2.41 - 16.8i)T^{2}
19 1+(4.57−5.27i)T+(−2.70−18.8i)T2 1 + (4.57 - 5.27i)T + (-2.70 - 18.8i)T^{2}
29 1+(−0.809−0.934i)T+(−4.12+28.7i)T2 1 + (-0.809 - 0.934i)T + (-4.12 + 28.7i)T^{2}
31 1+(0.611−2.08i)T+(−26.0−16.7i)T2 1 + (0.611 - 2.08i)T + (-26.0 - 16.7i)T^{2}
37 1+(−1.13−1.76i)T+(−15.3+33.6i)T2 1 + (-1.13 - 1.76i)T + (-15.3 + 33.6i)T^{2}
41 1+(1.45+0.937i)T+(17.0+37.2i)T2 1 + (1.45 + 0.937i)T + (17.0 + 37.2i)T^{2}
43 1+(−1.97+0.578i)T+(36.1−23.2i)T2 1 + (-1.97 + 0.578i)T + (36.1 - 23.2i)T^{2}
47 1−8.32iT−47T2 1 - 8.32iT - 47T^{2}
53 1+(−0.322+0.0464i)T+(50.8−14.9i)T2 1 + (-0.322 + 0.0464i)T + (50.8 - 14.9i)T^{2}
59 1+(−5.07−0.730i)T+(56.6+16.6i)T2 1 + (-5.07 - 0.730i)T + (56.6 + 16.6i)T^{2}
61 1+(−2.66+9.07i)T+(−51.3−32.9i)T2 1 + (-2.66 + 9.07i)T + (-51.3 - 32.9i)T^{2}
67 1+(−2.78−6.09i)T+(−43.8+50.6i)T2 1 + (-2.78 - 6.09i)T + (-43.8 + 50.6i)T^{2}
71 1+(1.45−0.663i)T+(46.4−53.6i)T2 1 + (1.45 - 0.663i)T + (46.4 - 53.6i)T^{2}
73 1+(−2.38+2.75i)T+(−10.3−72.2i)T2 1 + (-2.38 + 2.75i)T + (-10.3 - 72.2i)T^{2}
79 1+(−0.774+5.38i)T+(−75.7−22.2i)T2 1 + (-0.774 + 5.38i)T + (-75.7 - 22.2i)T^{2}
83 1+(−3.34+2.15i)T+(34.4−75.4i)T2 1 + (-3.34 + 2.15i)T + (34.4 - 75.4i)T^{2}
89 1+(2.20+7.50i)T+(−74.8+48.1i)T2 1 + (2.20 + 7.50i)T + (-74.8 + 48.1i)T^{2}
97 1+(0.772−1.20i)T+(−40.2−88.2i)T2 1 + (0.772 - 1.20i)T + (-40.2 - 88.2i)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

−14.40702145240037234050440405854, −13.11531542495242516488648688060, −12.55881718023319049808956888252, −10.88847416304262653711562548782, −10.31286441760135226011677586580, −8.090076720358971355399272341832, −7.18029446485271097771651512451, −6.41138801803906228999363086380, −4.32865237832904758823152003567, −3.44174039805917484702738599173, 2.26101115351629662376998210342, 4.29158992518998996451451783830, 5.26686916973626460989585716452, 6.74037719191339199653479305395, 8.697331014117982499615012384942, 9.455387229300003226278081356439, 11.16354894927142264315821887241, 11.89251018508664892926931795693, 12.81813550611695868185251295481, 13.54679911234859327530392380309

Graph of the ZZ-function along the critical line