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Properties

Label 2-380-19.4-c1-0-0
Degree 22
Conductor 380380
Sign −0.849−0.527i-0.849 - 0.527i
Analytic cond. 3.034313.03431
Root an. cond. 1.741921.74192
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  + (0.543 + 3.08i)3-s + (−0.766 + 0.642i)5-s + (0.0481 − 0.0833i)7-s + (−6.37 + 2.32i)9-s + (0.190 + 0.329i)11-s + (−0.668 + 3.79i)13-s + (−2.39 − 2.01i)15-s + (2.49 + 0.909i)17-s + (−0.788 − 4.28i)19-s + (0.283 + 0.103i)21-s + (0.131 + 0.110i)23-s + (0.173 − 0.984i)25-s + (−5.92 − 10.2i)27-s + (−3.67 + 1.33i)29-s + (−1.23 + 2.13i)31-s + ⋯
L(s)  = 1  + (0.313 + 1.77i)3-s + (−0.342 + 0.287i)5-s + (0.0181 − 0.0315i)7-s + (−2.12 + 0.773i)9-s + (0.0573 + 0.0994i)11-s + (−0.185 + 1.05i)13-s + (−0.618 − 0.519i)15-s + (0.605 + 0.220i)17-s + (−0.180 − 0.983i)19-s + (0.0617 + 0.0224i)21-s + (0.0273 + 0.0229i)23-s + (0.0347 − 0.196i)25-s + (−1.14 − 1.97i)27-s + (−0.682 + 0.248i)29-s + (−0.221 + 0.383i)31-s + ⋯

Functional equation

Λ(s)=(380s/2 ΓC(s) L(s)=((−0.849−0.527i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(380s/2 ΓC(s+1/2) L(s)=((−0.849−0.527i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 380380    =    22⋅5⋅192^{2} \cdot 5 \cdot 19
Sign: −0.849−0.527i-0.849 - 0.527i
Analytic conductor: 3.034313.03431
Root analytic conductor: 1.741921.74192
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ380(61,⋅)\chi_{380} (61, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 380, ( :1/2), −0.849−0.527i)(2,\ 380,\ (\ :1/2),\ -0.849 - 0.527i)

Particular Values

L(1)L(1) ≈\approx 0.336855+1.18222i0.336855 + 1.18222i
L(12)L(\frac12) ≈\approx 0.336855+1.18222i0.336855 + 1.18222i
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
5 1+(0.766−0.642i)T 1 + (0.766 - 0.642i)T
19 1+(0.788+4.28i)T 1 + (0.788 + 4.28i)T
good3 1+(−0.543−3.08i)T+(−2.81+1.02i)T2 1 + (-0.543 - 3.08i)T + (-2.81 + 1.02i)T^{2}
7 1+(−0.0481+0.0833i)T+(−3.5−6.06i)T2 1 + (-0.0481 + 0.0833i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.190−0.329i)T+(−5.5+9.52i)T2 1 + (-0.190 - 0.329i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.668−3.79i)T+(−12.2−4.44i)T2 1 + (0.668 - 3.79i)T + (-12.2 - 4.44i)T^{2}
17 1+(−2.49−0.909i)T+(13.0+10.9i)T2 1 + (-2.49 - 0.909i)T + (13.0 + 10.9i)T^{2}
23 1+(−0.131−0.110i)T+(3.99+22.6i)T2 1 + (-0.131 - 0.110i)T + (3.99 + 22.6i)T^{2}
29 1+(3.67−1.33i)T+(22.2−18.6i)T2 1 + (3.67 - 1.33i)T + (22.2 - 18.6i)T^{2}
31 1+(1.23−2.13i)T+(−15.5−26.8i)T2 1 + (1.23 - 2.13i)T + (-15.5 - 26.8i)T^{2}
37 1−6.92T+37T2 1 - 6.92T + 37T^{2}
41 1+(−0.688−3.90i)T+(−38.5+14.0i)T2 1 + (-0.688 - 3.90i)T + (-38.5 + 14.0i)T^{2}
43 1+(−9.81+8.23i)T+(7.46−42.3i)T2 1 + (-9.81 + 8.23i)T + (7.46 - 42.3i)T^{2}
47 1+(10.5−3.84i)T+(36.0−30.2i)T2 1 + (10.5 - 3.84i)T + (36.0 - 30.2i)T^{2}
53 1+(−9.54−8.00i)T+(9.20+52.1i)T2 1 + (-9.54 - 8.00i)T + (9.20 + 52.1i)T^{2}
59 1+(−5.79−2.10i)T+(45.1+37.9i)T2 1 + (-5.79 - 2.10i)T + (45.1 + 37.9i)T^{2}
61 1+(3.93+3.29i)T+(10.5+60.0i)T2 1 + (3.93 + 3.29i)T + (10.5 + 60.0i)T^{2}
67 1+(−11.4+4.16i)T+(51.3−43.0i)T2 1 + (-11.4 + 4.16i)T + (51.3 - 43.0i)T^{2}
71 1+(−1.30+1.09i)T+(12.3−69.9i)T2 1 + (-1.30 + 1.09i)T + (12.3 - 69.9i)T^{2}
73 1+(−0.258−1.46i)T+(−68.5+24.9i)T2 1 + (-0.258 - 1.46i)T + (-68.5 + 24.9i)T^{2}
79 1+(−2.60−14.7i)T+(−74.2+27.0i)T2 1 + (-2.60 - 14.7i)T + (-74.2 + 27.0i)T^{2}
83 1+(5.46−9.46i)T+(−41.5−71.8i)T2 1 + (5.46 - 9.46i)T + (-41.5 - 71.8i)T^{2}
89 1+(0.400−2.27i)T+(−83.6−30.4i)T2 1 + (0.400 - 2.27i)T + (-83.6 - 30.4i)T^{2}
97 1+(−1.37−0.500i)T+(74.3+62.3i)T2 1 + (-1.37 - 0.500i)T + (74.3 + 62.3i)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

−11.26040888284745147038267090852, −10.89208962977883610917943113645, −9.761342718364732059926155837284, −9.251356994718058513256970562404, −8.310672643408859551500710352140, −7.07942691082371814052565666666, −5.67987904475250700972176126559, −4.56866012931917474245087520321, −3.86920687271814072908267068209, −2.67843427951203647009597744529, 0.811045728740192393632785599695, 2.26912502090536573161364769167, 3.56093221549864961408204670418, 5.43509843503513700113854328340, 6.28225384825922054079542016160, 7.50686420506403422311708102424, 7.891978735408903071778627132677, 8.780489971601836749591880506044, 10.02417760585727650981445102649, 11.37097410144899956234132381674

Graph of the ZZ-function along the critical line