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Properties

Label 2-476-119.10-c1-0-2
Degree 22
Conductor 476476
Sign 0.690−0.723i0.690 - 0.723i
Analytic cond. 3.800873.80087
Root an. cond. 1.949581.94958
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  + (−0.110 − 0.223i)3-s + (−0.800 + 0.701i)5-s + (−1.28 + 2.31i)7-s + (1.78 − 2.33i)9-s + (3.58 − 0.235i)11-s + (1.36 + 1.36i)13-s + (0.245 + 0.101i)15-s + (−3.82 + 1.54i)17-s + (0.937 + 7.11i)19-s + (0.659 + 0.0324i)21-s + (7.25 + 3.57i)23-s + (−0.504 + 3.83i)25-s + (−1.45 − 0.289i)27-s + (0.397 + 1.99i)29-s + (5.57 − 2.75i)31-s + ⋯
L(s)  = 1  + (−0.0637 − 0.129i)3-s + (−0.357 + 0.313i)5-s + (−0.485 + 0.874i)7-s + (0.596 − 0.776i)9-s + (1.08 − 0.0709i)11-s + (0.378 + 0.378i)13-s + (0.0633 + 0.0262i)15-s + (−0.927 + 0.373i)17-s + (0.215 + 1.63i)19-s + (0.143 + 0.00708i)21-s + (1.51 + 0.745i)23-s + (−0.100 + 0.766i)25-s + (−0.279 − 0.0556i)27-s + (0.0737 + 0.370i)29-s + (1.00 − 0.494i)31-s + ⋯

Functional equation

Λ(s)=(476s/2 ΓC(s) L(s)=((0.690−0.723i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(476s/2 ΓC(s+1/2) L(s)=((0.690−0.723i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 476476    =    22⋅7⋅172^{2} \cdot 7 \cdot 17
Sign: 0.690−0.723i0.690 - 0.723i
Analytic conductor: 3.800873.80087
Root analytic conductor: 1.949581.94958
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ476(129,⋅)\chi_{476} (129, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 476, ( :1/2), 0.690−0.723i)(2,\ 476,\ (\ :1/2),\ 0.690 - 0.723i)

Particular Values

L(1)L(1) ≈\approx 1.20903+0.517610i1.20903 + 0.517610i
L(12)L(\frac12) ≈\approx 1.20903+0.517610i1.20903 + 0.517610i
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
7 1+(1.28−2.31i)T 1 + (1.28 - 2.31i)T
17 1+(3.82−1.54i)T 1 + (3.82 - 1.54i)T
good3 1+(0.110+0.223i)T+(−1.82+2.38i)T2 1 + (0.110 + 0.223i)T + (-1.82 + 2.38i)T^{2}
5 1+(0.800−0.701i)T+(0.652−4.95i)T2 1 + (0.800 - 0.701i)T + (0.652 - 4.95i)T^{2}
11 1+(−3.58+0.235i)T+(10.9−1.43i)T2 1 + (-3.58 + 0.235i)T + (10.9 - 1.43i)T^{2}
13 1+(−1.36−1.36i)T+13iT2 1 + (-1.36 - 1.36i)T + 13iT^{2}
19 1+(−0.937−7.11i)T+(−18.3+4.91i)T2 1 + (-0.937 - 7.11i)T + (-18.3 + 4.91i)T^{2}
23 1+(−7.25−3.57i)T+(14.0+18.2i)T2 1 + (-7.25 - 3.57i)T + (14.0 + 18.2i)T^{2}
29 1+(−0.397−1.99i)T+(−26.7+11.0i)T2 1 + (-0.397 - 1.99i)T + (-26.7 + 11.0i)T^{2}
31 1+(−5.57+2.75i)T+(18.8−24.5i)T2 1 + (-5.57 + 2.75i)T + (18.8 - 24.5i)T^{2}
37 1+(−0.117+1.78i)T+(−36.6−4.82i)T2 1 + (-0.117 + 1.78i)T + (-36.6 - 4.82i)T^{2}
41 1+(−0.719+3.61i)T+(−37.8−15.6i)T2 1 + (-0.719 + 3.61i)T + (-37.8 - 15.6i)T^{2}
43 1+(2.66+6.42i)T+(−30.4+30.4i)T2 1 + (2.66 + 6.42i)T + (-30.4 + 30.4i)T^{2}
47 1+(0.501−1.87i)T+(−40.7−23.5i)T2 1 + (0.501 - 1.87i)T + (-40.7 - 23.5i)T^{2}
53 1+(2.76+3.60i)T+(−13.7+51.1i)T2 1 + (2.76 + 3.60i)T + (-13.7 + 51.1i)T^{2}
59 1+(−9.46−1.24i)T+(56.9+15.2i)T2 1 + (-9.46 - 1.24i)T + (56.9 + 15.2i)T^{2}
61 1+(−3.71−10.9i)T+(−48.3+37.1i)T2 1 + (-3.71 - 10.9i)T + (-48.3 + 37.1i)T^{2}
67 1+(13.7+7.94i)T+(33.5+58.0i)T2 1 + (13.7 + 7.94i)T + (33.5 + 58.0i)T^{2}
71 1+(−7.66−5.11i)T+(27.1+65.5i)T2 1 + (-7.66 - 5.11i)T + (27.1 + 65.5i)T^{2}
73 1+(11.9+4.06i)T+(57.9+44.4i)T2 1 + (11.9 + 4.06i)T + (57.9 + 44.4i)T^{2}
79 1+(0.276−0.560i)T+(−48.0−62.6i)T2 1 + (0.276 - 0.560i)T + (-48.0 - 62.6i)T^{2}
83 1+(−2.77+6.70i)T+(−58.6−58.6i)T2 1 + (-2.77 + 6.70i)T + (-58.6 - 58.6i)T^{2}
89 1+(15.6+4.19i)T+(77.0+44.5i)T2 1 + (15.6 + 4.19i)T + (77.0 + 44.5i)T^{2}
97 1+(6.30−1.25i)T+(89.6−37.1i)T2 1 + (6.30 - 1.25i)T + (89.6 - 37.1i)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

−11.36964466797167433095669935590, −10.13675687536522977744650394208, −9.224644086938623996016274237865, −8.660900161596278919535925353365, −7.25452886391513091772533666873, −6.53384471540688302335839779817, −5.69260425735926310457065894642, −4.10482578954709387496280319227, −3.30446573680036534102643754024, −1.55840516018418995849788571932, 0.932144614813464166262289846418, 2.87053497384524878004656492189, 4.31521425242726446283332176463, 4.79022961757121759355867912279, 6.56628798445918636135106444522, 7.02362326169824470459969428249, 8.227821631814940014477524622250, 9.129327294997180727198497335192, 10.03158132459443081143344531259, 10.95165687055840715105450630566

Graph of the ZZ-function along the critical line