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Properties

Label 2-368-16.5-c1-0-35
Degree 22
Conductor 368368
Sign −0.978+0.206i-0.978 + 0.206i
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  + (−0.730 − 1.21i)2-s + (1.49 − 1.49i)3-s + (−0.932 + 1.76i)4-s + (−1.17 − 1.17i)5-s + (−2.89 − 0.717i)6-s − 0.836i·7-s + (2.82 − 0.163i)8-s − 1.46i·9-s + (−0.563 + 2.27i)10-s + (−2.72 − 2.72i)11-s + (1.24 + 4.03i)12-s + (3.86 − 3.86i)13-s + (−1.01 + 0.611i)14-s − 3.50·15-s + (−2.26 − 3.29i)16-s − 6.79·17-s + ⋯
L(s)  = 1  + (−0.516 − 0.856i)2-s + (0.862 − 0.862i)3-s + (−0.466 + 0.884i)4-s + (−0.524 − 0.524i)5-s + (−1.18 − 0.292i)6-s − 0.316i·7-s + (0.998 − 0.0577i)8-s − 0.487i·9-s + (−0.178 + 0.719i)10-s + (−0.823 − 0.823i)11-s + (0.360 + 1.16i)12-s + (1.07 − 1.07i)13-s + (−0.270 + 0.163i)14-s − 0.904·15-s + (−0.565 − 0.824i)16-s − 1.64·17-s + ⋯

Functional equation

Λ(s)=(368s/2 ΓC(s) L(s)=((−0.978+0.206i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(368s/2 ΓC(s+1/2) L(s)=((−0.978+0.206i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 368368    =    24⋅232^{4} \cdot 23
Sign: −0.978+0.206i-0.978 + 0.206i
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), −0.978+0.206i)(2,\ 368,\ (\ :1/2),\ -0.978 + 0.206i)

Particular Values

L(1)L(1) ≈\approx 0.107394−1.02929i0.107394 - 1.02929i
L(12)L(\frac12) ≈\approx 0.107394−1.02929i0.107394 - 1.02929i
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+(0.730+1.21i)T 1 + (0.730 + 1.21i)T
23 1−iT 1 - iT
good3 1+(−1.49+1.49i)T−3iT2 1 + (-1.49 + 1.49i)T - 3iT^{2}
5 1+(1.17+1.17i)T+5iT2 1 + (1.17 + 1.17i)T + 5iT^{2}
7 1+0.836iT−7T2 1 + 0.836iT - 7T^{2}
11 1+(2.72+2.72i)T+11iT2 1 + (2.72 + 2.72i)T + 11iT^{2}
13 1+(−3.86+3.86i)T−13iT2 1 + (-3.86 + 3.86i)T - 13iT^{2}
17 1+6.79T+17T2 1 + 6.79T + 17T^{2}
19 1+(2.46−2.46i)T−19iT2 1 + (2.46 - 2.46i)T - 19iT^{2}
29 1+(−3.58+3.58i)T−29iT2 1 + (-3.58 + 3.58i)T - 29iT^{2}
31 1−5.64T+31T2 1 - 5.64T + 31T^{2}
37 1+(−5.17−5.17i)T+37iT2 1 + (-5.17 - 5.17i)T + 37iT^{2}
41 1−2.36iT−41T2 1 - 2.36iT - 41T^{2}
43 1+(4.86+4.86i)T+43iT2 1 + (4.86 + 4.86i)T + 43iT^{2}
47 1+6.69T+47T2 1 + 6.69T + 47T^{2}
53 1+(−5.97−5.97i)T+53iT2 1 + (-5.97 - 5.97i)T + 53iT^{2}
59 1+(2.79+2.79i)T+59iT2 1 + (2.79 + 2.79i)T + 59iT^{2}
61 1+(−8.82+8.82i)T−61iT2 1 + (-8.82 + 8.82i)T - 61iT^{2}
67 1+(−0.659+0.659i)T−67iT2 1 + (-0.659 + 0.659i)T - 67iT^{2}
71 1+12.0iT−71T2 1 + 12.0iT - 71T^{2}
73 1−9.63iT−73T2 1 - 9.63iT - 73T^{2}
79 1+0.450T+79T2 1 + 0.450T + 79T^{2}
83 1+(−9.95+9.95i)T−83iT2 1 + (-9.95 + 9.95i)T - 83iT^{2}
89 1+4.51iT−89T2 1 + 4.51iT - 89T^{2}
97 1+2.31T+97T2 1 + 2.31T + 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

−10.93411607338609696268088148796, −10.21916016720917507940416899690, −8.719091260814787657327481540430, −8.300390448304282227589421167895, −7.82293505466857752503107930543, −6.41487598456945408092999703373, −4.64175895655977580855790592459, −3.41355724765996220278334269970, −2.33977441100508406934107123901, −0.76512604202615608171771781712, 2.39511395399818928211086739556, 4.03335339351287684093780284230, 4.77345448491554585162702474386, 6.40563439695196134229038413102, 7.13450759050434410048011342002, 8.441972792723812279691101916206, 8.831903277487998403909179895114, 9.765020118920476756972814873279, 10.67242122815063784638537701519, 11.42595771280400504049348432274

Graph of the ZZ-function along the critical line