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Properties

Label 2-176-11.10-c4-0-7
Degree 22
Conductor 176176
Sign 0.0167−0.999i0.0167 - 0.999i
Analytic cond. 18.193118.1931
Root an. cond. 4.265334.26533
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 11.2·3-s − 19.2·5-s + 33.9i·7-s + 45.7·9-s + (−2.03 + 120. i)11-s − 72.2i·13-s − 216.·15-s + 517. i·17-s + 411. i·19-s + 382. i·21-s + 967.·23-s − 254.·25-s − 396.·27-s − 640. i·29-s − 716.·31-s + ⋯
L(s)  = 1  + 1.25·3-s − 0.770·5-s + 0.692i·7-s + 0.564·9-s + (−0.0167 + 0.999i)11-s − 0.427i·13-s − 0.963·15-s + 1.79i·17-s + 1.14i·19-s + 0.866i·21-s + 1.82·23-s − 0.406·25-s − 0.544·27-s − 0.761i·29-s − 0.745·31-s + ⋯

Functional equation

Λ(s)=(176s/2 ΓC(s) L(s)=((0.0167−0.999i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0167 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(176s/2 ΓC(s+2) L(s)=((0.0167−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0167 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 176176    =    24⋅112^{4} \cdot 11
Sign: 0.0167−0.999i0.0167 - 0.999i
Analytic conductor: 18.193118.1931
Root analytic conductor: 4.265334.26533
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ176(65,⋅)\chi_{176} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 176, ( :2), 0.0167−0.999i)(2,\ 176,\ (\ :2),\ 0.0167 - 0.999i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 2.0526208292.052620829
L(12)L(\frac12) ≈\approx 2.0526208292.052620829
L(3)L(3) 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
11 1+(2.03−120.i)T 1 + (2.03 - 120. i)T
good3 1−11.2T+81T2 1 - 11.2T + 81T^{2}
5 1+19.2T+625T2 1 + 19.2T + 625T^{2}
7 1−33.9iT−2.40e3T2 1 - 33.9iT - 2.40e3T^{2}
13 1+72.2iT−2.85e4T2 1 + 72.2iT - 2.85e4T^{2}
17 1−517.iT−8.35e4T2 1 - 517. iT - 8.35e4T^{2}
19 1−411.iT−1.30e5T2 1 - 411. iT - 1.30e5T^{2}
23 1−967.T+2.79e5T2 1 - 967.T + 2.79e5T^{2}
29 1+640.iT−7.07e5T2 1 + 640. iT - 7.07e5T^{2}
31 1+716.T+9.23e5T2 1 + 716.T + 9.23e5T^{2}
37 1−1.20e3T+1.87e6T2 1 - 1.20e3T + 1.87e6T^{2}
41 1+966.iT−2.82e6T2 1 + 966. iT - 2.82e6T^{2}
43 1−2.72e3iT−3.41e6T2 1 - 2.72e3iT - 3.41e6T^{2}
47 1−177.T+4.87e6T2 1 - 177.T + 4.87e6T^{2}
53 1+4.02e3T+7.89e6T2 1 + 4.02e3T + 7.89e6T^{2}
59 1−1.98e3T+1.21e7T2 1 - 1.98e3T + 1.21e7T^{2}
61 1−2.39e3iT−1.38e7T2 1 - 2.39e3iT - 1.38e7T^{2}
67 1−2.00e3T+2.01e7T2 1 - 2.00e3T + 2.01e7T^{2}
71 1+4.25e3T+2.54e7T2 1 + 4.25e3T + 2.54e7T^{2}
73 1+1.28e3iT−2.83e7T2 1 + 1.28e3iT - 2.83e7T^{2}
79 1−1.48e3iT−3.89e7T2 1 - 1.48e3iT - 3.89e7T^{2}
83 1+1.01e4iT−4.74e7T2 1 + 1.01e4iT - 4.74e7T^{2}
89 1−5.13e3T+6.27e7T2 1 - 5.13e3T + 6.27e7T^{2}
97 1−2.23e3T+8.85e7T2 1 - 2.23e3T + 8.85e7T^{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

−12.52865114787149798963094615201, −11.33105758207708044305945615275, −10.08403211589838880826163166553, −9.047016428847625956627586145971, −8.172309303028481452628845598075, −7.50405042698457363415400353183, −5.90159629346529467564604498699, −4.25965013181730286989820552996, −3.19120030003018836432164258892, −1.85844272296792069921832938878, 0.65421129923708536656649908242, 2.75319924338731336957866415556, 3.63890038093974606249580929413, 4.97923637518562481977075944988, 6.96066361171237420373720509666, 7.64585889014894392290185240239, 8.813265512716154201510306838310, 9.369062176092942806076700413628, 10.97389110074375160764357886651, 11.56833360103969826122378403752

Graph of the ZZ-function along the critical line