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Properties

Label 2-368-16.5-c1-0-16
Degree 22
Conductor 368368
Sign 0.199−0.979i0.199 - 0.979i
Analytic cond. 2.938492.93849
Root an. cond. 1.714201.71420
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  + (0.0678 + 1.41i)2-s + (1.19 − 1.19i)3-s + (−1.99 + 0.191i)4-s + (0.672 + 0.672i)5-s + (1.77 + 1.60i)6-s + 1.79i·7-s + (−0.405 − 2.79i)8-s + 0.135i·9-s + (−0.903 + 0.994i)10-s + (2.49 + 2.49i)11-s + (−2.15 + 2.61i)12-s + (−0.0311 + 0.0311i)13-s + (−2.54 + 0.122i)14-s + 1.60·15-s + (3.92 − 0.762i)16-s + 4.24·17-s + ⋯
L(s)  = 1  + (0.0479 + 0.998i)2-s + (0.690 − 0.690i)3-s + (−0.995 + 0.0958i)4-s + (0.300 + 0.300i)5-s + (0.723 + 0.657i)6-s + 0.680i·7-s + (−0.143 − 0.989i)8-s + 0.0452i·9-s + (−0.285 + 0.314i)10-s + (0.753 + 0.753i)11-s + (−0.621 + 0.753i)12-s + (−0.00862 + 0.00862i)13-s + (−0.679 + 0.0326i)14-s + 0.415·15-s + (0.981 − 0.190i)16-s + 1.02·17-s + ⋯

Functional equation

Λ(s)=(368s/2 ΓC(s) L(s)=((0.199−0.979i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.199 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(368s/2 ΓC(s+1/2) L(s)=((0.199−0.979i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 368368    =    24⋅232^{4} \cdot 23
Sign: 0.199−0.979i0.199 - 0.979i
Analytic conductor: 2.938492.93849
Root analytic conductor: 1.714201.71420
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ368(277,⋅)\chi_{368} (277, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 368, ( :1/2), 0.199−0.979i)(2,\ 368,\ (\ :1/2),\ 0.199 - 0.979i)

Particular Values

L(1)L(1) ≈\approx 1.28382+1.04885i1.28382 + 1.04885i
L(12)L(\frac12) ≈\approx 1.28382+1.04885i1.28382 + 1.04885i
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+(−0.0678−1.41i)T 1 + (-0.0678 - 1.41i)T
23 1−iT 1 - iT
good3 1+(−1.19+1.19i)T−3iT2 1 + (-1.19 + 1.19i)T - 3iT^{2}
5 1+(−0.672−0.672i)T+5iT2 1 + (-0.672 - 0.672i)T + 5iT^{2}
7 1−1.79iT−7T2 1 - 1.79iT - 7T^{2}
11 1+(−2.49−2.49i)T+11iT2 1 + (-2.49 - 2.49i)T + 11iT^{2}
13 1+(0.0311−0.0311i)T−13iT2 1 + (0.0311 - 0.0311i)T - 13iT^{2}
17 1−4.24T+17T2 1 - 4.24T + 17T^{2}
19 1+(0.864−0.864i)T−19iT2 1 + (0.864 - 0.864i)T - 19iT^{2}
29 1+(1.05−1.05i)T−29iT2 1 + (1.05 - 1.05i)T - 29iT^{2}
31 1+2.27T+31T2 1 + 2.27T + 31T^{2}
37 1+(1.42+1.42i)T+37iT2 1 + (1.42 + 1.42i)T + 37iT^{2}
41 1+8.94iT−41T2 1 + 8.94iT - 41T^{2}
43 1+(2.22+2.22i)T+43iT2 1 + (2.22 + 2.22i)T + 43iT^{2}
47 1+9.40T+47T2 1 + 9.40T + 47T^{2}
53 1+(4.89+4.89i)T+53iT2 1 + (4.89 + 4.89i)T + 53iT^{2}
59 1+(6.23+6.23i)T+59iT2 1 + (6.23 + 6.23i)T + 59iT^{2}
61 1+(−0.841+0.841i)T−61iT2 1 + (-0.841 + 0.841i)T - 61iT^{2}
67 1+(6.47−6.47i)T−67iT2 1 + (6.47 - 6.47i)T - 67iT^{2}
71 1+6.62iT−71T2 1 + 6.62iT - 71T^{2}
73 1−0.712iT−73T2 1 - 0.712iT - 73T^{2}
79 1−3.84T+79T2 1 - 3.84T + 79T^{2}
83 1+(−7.62+7.62i)T−83iT2 1 + (-7.62 + 7.62i)T - 83iT^{2}
89 1+5.80iT−89T2 1 + 5.80iT - 89T^{2}
97 1−19.5T+97T2 1 - 19.5T + 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

−12.02230366521081108296006123371, −10.39267735417888742822590002843, −9.438106423335808905963154272691, −8.637664998458569544234143550505, −7.75577752214399301138333055050, −6.97731692383333059481503328111, −6.03131221159274706970645825924, −4.93732226625301715783472475202, −3.45007524292191615862839947186, −1.89793299546158056440045377543, 1.24237843368814889930926872295, 3.09173358998269879976215077889, 3.81736409660597829717095700935, 4.87306541795887202561496966457, 6.19164318771774917535962864101, 7.84364334170134135350181184829, 8.839656851146901821863345447091, 9.476110109082315564893212242121, 10.18803114990468966862578101662, 11.11990163907156563569599653494

Graph of the ZZ-function along the critical line