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Properties

Label 2-176-44.43-c5-0-12
Degree 22
Conductor 176176
Sign 0.723−0.690i0.723 - 0.690i
Analytic cond. 28.227528.2275
Root an. cond. 5.312965.31296
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.663i·3-s + 68.6·5-s − 214.·7-s + 242.·9-s + (95.0 + 389. i)11-s − 556. i·13-s − 45.5i·15-s + 1.84e3i·17-s + 1.21e3·19-s + 142. i·21-s + 121. i·23-s + 1.59e3·25-s − 322. i·27-s − 4.26e3i·29-s + 3.81e3i·31-s + ⋯
L(s)  = 1  − 0.0425i·3-s + 1.22·5-s − 1.65·7-s + 0.998·9-s + (0.236 + 0.971i)11-s − 0.912i·13-s − 0.0523i·15-s + 1.55i·17-s + 0.772·19-s + 0.0704i·21-s + 0.0478i·23-s + 0.509·25-s − 0.0850i·27-s − 0.942i·29-s + 0.712i·31-s + ⋯

Functional equation

Λ(s)=(176s/2 ΓC(s) L(s)=((0.723−0.690i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(176s/2 ΓC(s+5/2) L(s)=((0.723−0.690i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 176176    =    24⋅112^{4} \cdot 11
Sign: 0.723−0.690i0.723 - 0.690i
Analytic conductor: 28.227528.2275
Root analytic conductor: 5.312965.31296
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ176(175,⋅)\chi_{176} (175, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 176, ( :5/2), 0.723−0.690i)(2,\ 176,\ (\ :5/2),\ 0.723 - 0.690i)

Particular Values

L(3)L(3) ≈\approx 2.1519799602.151979960
L(12)L(\frac12) ≈\approx 2.1519799602.151979960
L(72)L(\frac{7}{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
11 1+(−95.0−389.i)T 1 + (-95.0 - 389. i)T
good3 1+0.663iT−243T2 1 + 0.663iT - 243T^{2}
5 1−68.6T+3.12e3T2 1 - 68.6T + 3.12e3T^{2}
7 1+214.T+1.68e4T2 1 + 214.T + 1.68e4T^{2}
13 1+556.iT−3.71e5T2 1 + 556. iT - 3.71e5T^{2}
17 1−1.84e3iT−1.41e6T2 1 - 1.84e3iT - 1.41e6T^{2}
19 1−1.21e3T+2.47e6T2 1 - 1.21e3T + 2.47e6T^{2}
23 1−121.iT−6.43e6T2 1 - 121. iT - 6.43e6T^{2}
29 1+4.26e3iT−2.05e7T2 1 + 4.26e3iT - 2.05e7T^{2}
31 1−3.81e3iT−2.86e7T2 1 - 3.81e3iT - 2.86e7T^{2}
37 1−9.38e3T+6.93e7T2 1 - 9.38e3T + 6.93e7T^{2}
41 1−1.48e4iT−1.15e8T2 1 - 1.48e4iT - 1.15e8T^{2}
43 1−1.82e4T+1.47e8T2 1 - 1.82e4T + 1.47e8T^{2}
47 1−2.42e4iT−2.29e8T2 1 - 2.42e4iT - 2.29e8T^{2}
53 1−1.80e4T+4.18e8T2 1 - 1.80e4T + 4.18e8T^{2}
59 1+970.iT−7.14e8T2 1 + 970. iT - 7.14e8T^{2}
61 1−4.83e4iT−8.44e8T2 1 - 4.83e4iT - 8.44e8T^{2}
67 1−9.34e3iT−1.35e9T2 1 - 9.34e3iT - 1.35e9T^{2}
71 1+4.24e4iT−1.80e9T2 1 + 4.24e4iT - 1.80e9T^{2}
73 1+2.18e3iT−2.07e9T2 1 + 2.18e3iT - 2.07e9T^{2}
79 1−4.97e4T+3.07e9T2 1 - 4.97e4T + 3.07e9T^{2}
83 1+4.08e4T+3.93e9T2 1 + 4.08e4T + 3.93e9T^{2}
89 1+4.75e3T+5.58e9T2 1 + 4.75e3T + 5.58e9T^{2}
97 1−4.44e4T+8.58e9T2 1 - 4.44e4T + 8.58e9T^{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

−12.37222787874297826681487083481, −10.48892895811698185839813721508, −9.867558249079231384078613230019, −9.357261206384853190007109919200, −7.64671048755687421520432482533, −6.48159079321768427836222827957, −5.80623189066641751433423374244, −4.14392379136184272921233457432, −2.70274337192125060369002126618, −1.26312776975819096148539061327, 0.77028841425878082681259496750, 2.44460162197206608749523141796, 3.73981780674209125648292559110, 5.40722749908951826122157837264, 6.43984065025531748247627099943, 7.18981385727520701006275069507, 9.274600281356355300127286306604, 9.425438218928426285586197225683, 10.43233470425463565050895554411, 11.76690275775265621097700447379

Graph of the ZZ-function along the critical line