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Properties

Label 2-368-16.5-c1-0-36
Degree 22
Conductor 368368
Sign 0.874+0.485i0.874 + 0.485i
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  + (1.29 + 0.578i)2-s + (1.73 − 1.73i)3-s + (1.33 + 1.49i)4-s + (−1.86 − 1.86i)5-s + (3.23 − 1.23i)6-s − 1.07i·7-s + (0.855 + 2.69i)8-s − 2.98i·9-s + (−1.33 − 3.49i)10-s + (0.551 + 0.551i)11-s + (4.88 + 0.278i)12-s + (0.244 − 0.244i)13-s + (0.621 − 1.38i)14-s − 6.46·15-s + (−0.454 + 3.97i)16-s + 3.95·17-s + ⋯
L(s)  = 1  + (0.912 + 0.408i)2-s + (0.998 − 0.998i)3-s + (0.665 + 0.746i)4-s + (−0.835 − 0.835i)5-s + (1.31 − 0.503i)6-s − 0.406i·7-s + (0.302 + 0.953i)8-s − 0.995i·9-s + (−0.421 − 1.10i)10-s + (0.166 + 0.166i)11-s + (1.41 + 0.0803i)12-s + (0.0676 − 0.0676i)13-s + (0.166 − 0.370i)14-s − 1.66·15-s + (−0.113 + 0.993i)16-s + 0.958·17-s + ⋯

Functional equation

Λ(s)=(368s/2 ΓC(s) L(s)=((0.874+0.485i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(368s/2 ΓC(s+1/2) L(s)=((0.874+0.485i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 368368    =    24⋅232^{4} \cdot 23
Sign: 0.874+0.485i0.874 + 0.485i
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), 0.874+0.485i)(2,\ 368,\ (\ :1/2),\ 0.874 + 0.485i)

Particular Values

L(1)L(1) ≈\approx 2.61906−0.677949i2.61906 - 0.677949i
L(12)L(\frac12) ≈\approx 2.61906−0.677949i2.61906 - 0.677949i
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+(−1.29−0.578i)T 1 + (-1.29 - 0.578i)T
23 1−iT 1 - iT
good3 1+(−1.73+1.73i)T−3iT2 1 + (-1.73 + 1.73i)T - 3iT^{2}
5 1+(1.86+1.86i)T+5iT2 1 + (1.86 + 1.86i)T + 5iT^{2}
7 1+1.07iT−7T2 1 + 1.07iT - 7T^{2}
11 1+(−0.551−0.551i)T+11iT2 1 + (-0.551 - 0.551i)T + 11iT^{2}
13 1+(−0.244+0.244i)T−13iT2 1 + (-0.244 + 0.244i)T - 13iT^{2}
17 1−3.95T+17T2 1 - 3.95T + 17T^{2}
19 1+(2.75−2.75i)T−19iT2 1 + (2.75 - 2.75i)T - 19iT^{2}
29 1+(4.25−4.25i)T−29iT2 1 + (4.25 - 4.25i)T - 29iT^{2}
31 1+2.33T+31T2 1 + 2.33T + 31T^{2}
37 1+(3.31+3.31i)T+37iT2 1 + (3.31 + 3.31i)T + 37iT^{2}
41 1−12.3iT−41T2 1 - 12.3iT - 41T^{2}
43 1+(8.16+8.16i)T+43iT2 1 + (8.16 + 8.16i)T + 43iT^{2}
47 1−4.08T+47T2 1 - 4.08T + 47T^{2}
53 1+(0.106+0.106i)T+53iT2 1 + (0.106 + 0.106i)T + 53iT^{2}
59 1+(−4.32−4.32i)T+59iT2 1 + (-4.32 - 4.32i)T + 59iT^{2}
61 1+(6.39−6.39i)T−61iT2 1 + (6.39 - 6.39i)T - 61iT^{2}
67 1+(−3.60+3.60i)T−67iT2 1 + (-3.60 + 3.60i)T - 67iT^{2}
71 1+10.4iT−71T2 1 + 10.4iT - 71T^{2}
73 1+6.16iT−73T2 1 + 6.16iT - 73T^{2}
79 1−1.42T+79T2 1 - 1.42T + 79T^{2}
83 1+(−4.07+4.07i)T−83iT2 1 + (-4.07 + 4.07i)T - 83iT^{2}
89 1−9.40iT−89T2 1 - 9.40iT - 89T^{2}
97 1−3.27T+97T2 1 - 3.27T + 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

−11.93529625786628450076115924882, −10.62435062296244368061705622020, −9.006729456468286289542942586833, −8.146776695650649158322614312436, −7.63128574783214032735802959407, −6.78810429184572225064337265093, −5.42531955924970885872087722242, −4.15538647809956733942896790948, −3.25722020181861591151610196326, −1.66368358065988289861878393599, 2.46409238528425893484450480780, 3.45202737014210836014899898999, 4.04828664753916538799321924118, 5.30803494397197178751393828404, 6.63984815454228368133162742588, 7.72911690080378929714869152598, 8.865295899851648455190663147565, 9.846524486673896234426364847271, 10.65520891812388269672909710833, 11.43192635605095202040183707277

Graph of the ZZ-function along the critical line