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Properties

Label 2-476-119.10-c1-0-8
Degree 22
Conductor 476476
Sign −0.221+0.975i-0.221 + 0.975i
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  + (−0.951 − 1.92i)3-s + (3.19 − 2.79i)5-s + (1.81 − 1.92i)7-s + (−0.991 + 1.29i)9-s + (1.13 − 0.0742i)11-s + (3.38 + 3.38i)13-s + (−8.43 − 3.49i)15-s + (−1.61 + 3.79i)17-s + (0.342 + 2.59i)19-s + (−5.44 − 1.65i)21-s + (−1.44 − 0.714i)23-s + (1.69 − 12.8i)25-s + (−2.89 − 0.575i)27-s + (1.39 + 7.02i)29-s + (−2.18 + 1.07i)31-s + ⋯
L(s)  = 1  + (−0.549 − 1.11i)3-s + (1.42 − 1.25i)5-s + (0.684 − 0.728i)7-s + (−0.330 + 0.430i)9-s + (0.341 − 0.0223i)11-s + (0.937 + 0.937i)13-s + (−2.17 − 0.902i)15-s + (−0.391 + 0.920i)17-s + (0.0785 + 0.596i)19-s + (−1.18 − 0.362i)21-s + (−0.302 − 0.148i)23-s + (0.339 − 2.57i)25-s + (−0.557 − 0.110i)27-s + (0.259 + 1.30i)29-s + (−0.392 + 0.193i)31-s + ⋯

Functional equation

Λ(s)=(476s/2 ΓC(s) L(s)=((−0.221+0.975i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(476s/2 ΓC(s+1/2) L(s)=((−0.221+0.975i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 476476    =    22⋅7⋅172^{2} \cdot 7 \cdot 17
Sign: −0.221+0.975i-0.221 + 0.975i
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), −0.221+0.975i)(2,\ 476,\ (\ :1/2),\ -0.221 + 0.975i)

Particular Values

L(1)L(1) ≈\approx 1.03387−1.29475i1.03387 - 1.29475i
L(12)L(\frac12) ≈\approx 1.03387−1.29475i1.03387 - 1.29475i
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+(−1.81+1.92i)T 1 + (-1.81 + 1.92i)T
17 1+(1.61−3.79i)T 1 + (1.61 - 3.79i)T
good3 1+(0.951+1.92i)T+(−1.82+2.38i)T2 1 + (0.951 + 1.92i)T + (-1.82 + 2.38i)T^{2}
5 1+(−3.19+2.79i)T+(0.652−4.95i)T2 1 + (-3.19 + 2.79i)T + (0.652 - 4.95i)T^{2}
11 1+(−1.13+0.0742i)T+(10.9−1.43i)T2 1 + (-1.13 + 0.0742i)T + (10.9 - 1.43i)T^{2}
13 1+(−3.38−3.38i)T+13iT2 1 + (-3.38 - 3.38i)T + 13iT^{2}
19 1+(−0.342−2.59i)T+(−18.3+4.91i)T2 1 + (-0.342 - 2.59i)T + (-18.3 + 4.91i)T^{2}
23 1+(1.44+0.714i)T+(14.0+18.2i)T2 1 + (1.44 + 0.714i)T + (14.0 + 18.2i)T^{2}
29 1+(−1.39−7.02i)T+(−26.7+11.0i)T2 1 + (-1.39 - 7.02i)T + (-26.7 + 11.0i)T^{2}
31 1+(2.18−1.07i)T+(18.8−24.5i)T2 1 + (2.18 - 1.07i)T + (18.8 - 24.5i)T^{2}
37 1+(0.312−4.76i)T+(−36.6−4.82i)T2 1 + (0.312 - 4.76i)T + (-36.6 - 4.82i)T^{2}
41 1+(1.07−5.39i)T+(−37.8−15.6i)T2 1 + (1.07 - 5.39i)T + (-37.8 - 15.6i)T^{2}
43 1+(2.41+5.83i)T+(−30.4+30.4i)T2 1 + (2.41 + 5.83i)T + (-30.4 + 30.4i)T^{2}
47 1+(1.67−6.23i)T+(−40.7−23.5i)T2 1 + (1.67 - 6.23i)T + (-40.7 - 23.5i)T^{2}
53 1+(6.29+8.20i)T+(−13.7+51.1i)T2 1 + (6.29 + 8.20i)T + (-13.7 + 51.1i)T^{2}
59 1+(6.62+0.872i)T+(56.9+15.2i)T2 1 + (6.62 + 0.872i)T + (56.9 + 15.2i)T^{2}
61 1+(−3.08−9.08i)T+(−48.3+37.1i)T2 1 + (-3.08 - 9.08i)T + (-48.3 + 37.1i)T^{2}
67 1+(4.42+2.55i)T+(33.5+58.0i)T2 1 + (4.42 + 2.55i)T + (33.5 + 58.0i)T^{2}
71 1+(−2.82−1.89i)T+(27.1+65.5i)T2 1 + (-2.82 - 1.89i)T + (27.1 + 65.5i)T^{2}
73 1+(−1.40−0.476i)T+(57.9+44.4i)T2 1 + (-1.40 - 0.476i)T + (57.9 + 44.4i)T^{2}
79 1+(5.29−10.7i)T+(−48.0−62.6i)T2 1 + (5.29 - 10.7i)T + (-48.0 - 62.6i)T^{2}
83 1+(5.81−14.0i)T+(−58.6−58.6i)T2 1 + (5.81 - 14.0i)T + (-58.6 - 58.6i)T^{2}
89 1+(1.50+0.403i)T+(77.0+44.5i)T2 1 + (1.50 + 0.403i)T + (77.0 + 44.5i)T^{2}
97 1+(−15.8+3.15i)T+(89.6−37.1i)T2 1 + (-15.8 + 3.15i)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

−10.83534534848884409857042871445, −9.844550757755530599124197854166, −8.817547348276293128728097699391, −8.146937502095732262371991065826, −6.75929950331032039693160320350, −6.20610440821376850263558685083, −5.22485734034695650459207774104, −4.13339134494885328303927366636, −1.63005566369249592255550265378, −1.39585857736458637107098392478, 2.10213942323596683243730825282, 3.27460062719398352302284290135, 4.76487929964109958125855460275, 5.69557858812245686069530625679, 6.22365689850062963081340331035, 7.53042991190406721863168586625, 8.968336404646523615012669332295, 9.638926467916073160323376236064, 10.44708672403342955259373694063, 11.04937692913201771272948031327

Graph of the ZZ-function along the critical line