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Properties

Label 2-152-152.75-c1-0-13
Degree 22
Conductor 152152
Sign −0.133+0.991i-0.133 + 0.991i
Analytic cond. 1.213721.21372
Root an. cond. 1.101691.10169
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  + (0.491 − 1.32i)2-s + 0.754i·3-s + (−1.51 − 1.30i)4-s − 2.08i·5-s + (1 + 0.370i)6-s − 2.23i·7-s + (−2.47 + 1.37i)8-s + 2.43·9-s + (−2.76 − 1.02i)10-s − 0.602·11-s + (0.982 − 1.14i)12-s + 1.29·13-s + (−2.96 − 1.09i)14-s + 1.57·15-s + (0.602 + 3.95i)16-s + 2.20·17-s + ⋯
L(s)  = 1  + (0.347 − 0.937i)2-s + 0.435i·3-s + (−0.758 − 0.651i)4-s − 0.934i·5-s + (0.408 + 0.151i)6-s − 0.845i·7-s + (−0.874 + 0.484i)8-s + 0.810·9-s + (−0.875 − 0.324i)10-s − 0.181·11-s + (0.283 − 0.330i)12-s + 0.359·13-s + (−0.792 − 0.293i)14-s + 0.406·15-s + (0.150 + 0.988i)16-s + 0.534·17-s + ⋯

Functional equation

Λ(s)=(152s/2 ΓC(s) L(s)=((−0.133+0.991i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(152s/2 ΓC(s+1/2) L(s)=((−0.133+0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 152152    =    23⋅192^{3} \cdot 19
Sign: −0.133+0.991i-0.133 + 0.991i
Analytic conductor: 1.213721.21372
Root analytic conductor: 1.101691.10169
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ152(75,⋅)\chi_{152} (75, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 152, ( :1/2), −0.133+0.991i)(2,\ 152,\ (\ :1/2),\ -0.133 + 0.991i)

Particular Values

L(1)L(1) ≈\approx 0.819397−0.937028i0.819397 - 0.937028i
L(12)L(\frac12) ≈\approx 0.819397−0.937028i0.819397 - 0.937028i
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+(−0.491+1.32i)T 1 + (-0.491 + 1.32i)T
19 1+(2.60−3.49i)T 1 + (2.60 - 3.49i)T
good3 1−0.754iT−3T2 1 - 0.754iT - 3T^{2}
5 1+2.08iT−5T2 1 + 2.08iT - 5T^{2}
7 1+2.23iT−7T2 1 + 2.23iT - 7T^{2}
11 1+0.602T+11T2 1 + 0.602T + 11T^{2}
13 1−1.29T+13T2 1 - 1.29T + 13T^{2}
17 1−2.20T+17T2 1 - 2.20T + 17T^{2}
23 1−6.19iT−23T2 1 - 6.19iT - 23T^{2}
29 1+8.20T+29T2 1 + 8.20T + 29T^{2}
31 1−4.94T+31T2 1 - 4.94T + 31T^{2}
37 1−6.91T+37T2 1 - 6.91T + 37T^{2}
41 1+6.53iT−41T2 1 + 6.53iT - 41T^{2}
43 1+0.191T+43T2 1 + 0.191T + 43T^{2}
47 1−0.223iT−47T2 1 - 0.223iT - 47T^{2}
53 1+4.83T+53T2 1 + 4.83T + 53T^{2}
59 1+5.60iT−59T2 1 + 5.60iT - 59T^{2}
61 1−11.7iT−61T2 1 - 11.7iT - 61T^{2}
67 1+6.23iT−67T2 1 + 6.23iT - 67T^{2}
71 1+12.4T+71T2 1 + 12.4T + 71T^{2}
73 1+8.27T+73T2 1 + 8.27T + 73T^{2}
79 1+15.4T+79T2 1 + 15.4T + 79T^{2}
83 1−2T+83T2 1 - 2T + 83T^{2}
89 1+7.44iT−89T2 1 + 7.44iT - 89T^{2}
97 1−17.6iT−97T2 1 - 17.6iT - 97T^{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

−12.86467238139947487800576822675, −11.74987986945764513651194017499, −10.62619775345373095870098705051, −9.892525981430369545325916941861, −8.952337539001952514743487742608, −7.60493299219672069042240902268, −5.73733287039432734139483405612, −4.53068082412825169969255610087, −3.68925899749569067221896621208, −1.37884138731319244440393907220, 2.79293148298529568667643952077, 4.46323300962751062202782564014, 5.97824227559216108471179333755, 6.78473533674838398708409798546, 7.76863075865256490109849373093, 8.882596223234786853778538277567, 10.11336723477964934869521665060, 11.44660812589026750076984440729, 12.63938654234220350820687296576, 13.22469991580347004574451578416

Graph of the ZZ-function along the critical line