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Properties

Label 2-176-16.13-c1-0-8
Degree 22
Conductor 176176
Sign 0.506−0.862i0.506 - 0.862i
Analytic cond. 1.405361.40536
Root an. cond. 1.185481.18548
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  + (1.28 + 0.586i)2-s + (0.586 + 0.586i)3-s + (1.31 + 1.50i)4-s + (−1.32 + 1.32i)5-s + (0.411 + 1.09i)6-s − 1.42i·7-s + (0.805 + 2.71i)8-s − 2.31i·9-s + (−2.47 + 0.926i)10-s + (−0.707 + 0.707i)11-s + (−0.114 + 1.65i)12-s + (−1.74 − 1.74i)13-s + (0.832 − 1.82i)14-s − 1.54·15-s + (−0.552 + 3.96i)16-s + 6.99·17-s + ⋯
L(s)  = 1  + (0.910 + 0.414i)2-s + (0.338 + 0.338i)3-s + (0.656 + 0.754i)4-s + (−0.590 + 0.590i)5-s + (0.167 + 0.448i)6-s − 0.537i·7-s + (0.284 + 0.958i)8-s − 0.770i·9-s + (−0.782 + 0.292i)10-s + (−0.213 + 0.213i)11-s + (−0.0331 + 0.477i)12-s + (−0.482 − 0.482i)13-s + (0.222 − 0.488i)14-s − 0.400·15-s + (−0.138 + 0.990i)16-s + 1.69·17-s + ⋯

Functional equation

Λ(s)=(176s/2 ΓC(s) L(s)=((0.506−0.862i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(176s/2 ΓC(s+1/2) L(s)=((0.506−0.862i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 176176    =    24⋅112^{4} \cdot 11
Sign: 0.506−0.862i0.506 - 0.862i
Analytic conductor: 1.405361.40536
Root analytic conductor: 1.185481.18548
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ176(45,⋅)\chi_{176} (45, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 176, ( :1/2), 0.506−0.862i)(2,\ 176,\ (\ :1/2),\ 0.506 - 0.862i)

Particular Values

L(1)L(1) ≈\approx 1.64410+0.940891i1.64410 + 0.940891i
L(12)L(\frac12) ≈\approx 1.64410+0.940891i1.64410 + 0.940891i
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+(−1.28−0.586i)T 1 + (-1.28 - 0.586i)T
11 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good3 1+(−0.586−0.586i)T+3iT2 1 + (-0.586 - 0.586i)T + 3iT^{2}
5 1+(1.32−1.32i)T−5iT2 1 + (1.32 - 1.32i)T - 5iT^{2}
7 1+1.42iT−7T2 1 + 1.42iT - 7T^{2}
13 1+(1.74+1.74i)T+13iT2 1 + (1.74 + 1.74i)T + 13iT^{2}
17 1−6.99T+17T2 1 - 6.99T + 17T^{2}
19 1+(3.48+3.48i)T+19iT2 1 + (3.48 + 3.48i)T + 19iT^{2}
23 1+4.79iT−23T2 1 + 4.79iT - 23T^{2}
29 1+(−3.38−3.38i)T+29iT2 1 + (-3.38 - 3.38i)T + 29iT^{2}
31 1+8.04T+31T2 1 + 8.04T + 31T^{2}
37 1+(4.09−4.09i)T−37iT2 1 + (4.09 - 4.09i)T - 37iT^{2}
41 1+5.85iT−41T2 1 + 5.85iT - 41T^{2}
43 1+(3.09−3.09i)T−43iT2 1 + (3.09 - 3.09i)T - 43iT^{2}
47 1+8.12T+47T2 1 + 8.12T + 47T^{2}
53 1+(−0.759+0.759i)T−53iT2 1 + (-0.759 + 0.759i)T - 53iT^{2}
59 1+(0.513−0.513i)T−59iT2 1 + (0.513 - 0.513i)T - 59iT^{2}
61 1+(−0.211−0.211i)T+61iT2 1 + (-0.211 - 0.211i)T + 61iT^{2}
67 1+(−0.116−0.116i)T+67iT2 1 + (-0.116 - 0.116i)T + 67iT^{2}
71 1−9.46iT−71T2 1 - 9.46iT - 71T^{2}
73 1−7.55iT−73T2 1 - 7.55iT - 73T^{2}
79 1−8.55T+79T2 1 - 8.55T + 79T^{2}
83 1+(−7.36−7.36i)T+83iT2 1 + (-7.36 - 7.36i)T + 83iT^{2}
89 1+2.68iT−89T2 1 + 2.68iT - 89T^{2}
97 1+15.2T+97T2 1 + 15.2T + 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

−12.80747907346369996469753564601, −12.15534748108069616514181036369, −10.98985508231822278945888708155, −10.07628587502780602349513345552, −8.543911677762268319955564594942, −7.44989553161865205178339994401, −6.67251551804783985519901297433, −5.17890179523264759313165165811, −3.87076134641849719957577952545, −2.99754421876959378699347313631, 1.90901546532702170384354139076, 3.46174778020870897351321714803, 4.84743254656332924167516206599, 5.82379556150558926564632219376, 7.40319265247117702815011598845, 8.280779018005384455581630284630, 9.691577621744629240216444275942, 10.78893188738063612379133266093, 11.99493681703418838648522486766, 12.40388000193432887267715346599

Graph of the ZZ-function along the critical line