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Properties

Label 2-368-16.13-c1-0-40
Degree 22
Conductor 368368
Sign −0.968+0.249i-0.968 + 0.249i
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  + (0.586 − 1.28i)2-s + (−0.955 − 0.955i)3-s + (−1.31 − 1.51i)4-s + (2.38 − 2.38i)5-s + (−1.79 + 0.668i)6-s + 0.198i·7-s + (−2.71 + 0.801i)8-s − 1.17i·9-s + (−1.66 − 4.46i)10-s + (−0.110 + 0.110i)11-s + (−0.189 + 2.69i)12-s + (1.69 + 1.69i)13-s + (0.255 + 0.116i)14-s − 4.55·15-s + (−0.560 + 3.96i)16-s − 0.645·17-s + ⋯
L(s)  = 1  + (0.414 − 0.909i)2-s + (−0.551 − 0.551i)3-s + (−0.655 − 0.755i)4-s + (1.06 − 1.06i)5-s + (−0.730 + 0.273i)6-s + 0.0751i·7-s + (−0.959 + 0.283i)8-s − 0.391i·9-s + (−0.527 − 1.41i)10-s + (−0.0332 + 0.0332i)11-s + (−0.0548 + 0.778i)12-s + (0.469 + 0.469i)13-s + (0.0683 + 0.0311i)14-s − 1.17·15-s + (−0.140 + 0.990i)16-s − 0.156·17-s + ⋯

Functional equation

Λ(s)=(368s/2 ΓC(s) L(s)=((−0.968+0.249i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(368s/2 ΓC(s+1/2) L(s)=((−0.968+0.249i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 368368    =    24⋅232^{4} \cdot 23
Sign: −0.968+0.249i-0.968 + 0.249i
Analytic conductor: 2.938492.93849
Root analytic conductor: 1.714201.71420
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ368(93,⋅)\chi_{368} (93, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 368, ( :1/2), −0.968+0.249i)(2,\ 368,\ (\ :1/2),\ -0.968 + 0.249i)

Particular Values

L(1)L(1) ≈\approx 0.180764−1.42656i0.180764 - 1.42656i
L(12)L(\frac12) ≈\approx 0.180764−1.42656i0.180764 - 1.42656i
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+(−0.586+1.28i)T 1 + (-0.586 + 1.28i)T
23 1+iT 1 + iT
good3 1+(0.955+0.955i)T+3iT2 1 + (0.955 + 0.955i)T + 3iT^{2}
5 1+(−2.38+2.38i)T−5iT2 1 + (-2.38 + 2.38i)T - 5iT^{2}
7 1−0.198iT−7T2 1 - 0.198iT - 7T^{2}
11 1+(0.110−0.110i)T−11iT2 1 + (0.110 - 0.110i)T - 11iT^{2}
13 1+(−1.69−1.69i)T+13iT2 1 + (-1.69 - 1.69i)T + 13iT^{2}
17 1+0.645T+17T2 1 + 0.645T + 17T^{2}
19 1+(−0.173−0.173i)T+19iT2 1 + (-0.173 - 0.173i)T + 19iT^{2}
29 1+(−2.98−2.98i)T+29iT2 1 + (-2.98 - 2.98i)T + 29iT^{2}
31 1+4.74T+31T2 1 + 4.74T + 31T^{2}
37 1+(1.53−1.53i)T−37iT2 1 + (1.53 - 1.53i)T - 37iT^{2}
41 1+9.42iT−41T2 1 + 9.42iT - 41T^{2}
43 1+(2.46−2.46i)T−43iT2 1 + (2.46 - 2.46i)T - 43iT^{2}
47 1−10.2T+47T2 1 - 10.2T + 47T^{2}
53 1+(−6.13+6.13i)T−53iT2 1 + (-6.13 + 6.13i)T - 53iT^{2}
59 1+(−3.18+3.18i)T−59iT2 1 + (-3.18 + 3.18i)T - 59iT^{2}
61 1+(−0.134−0.134i)T+61iT2 1 + (-0.134 - 0.134i)T + 61iT^{2}
67 1+(−8.34−8.34i)T+67iT2 1 + (-8.34 - 8.34i)T + 67iT^{2}
71 1+4.76iT−71T2 1 + 4.76iT - 71T^{2}
73 1+5.99iT−73T2 1 + 5.99iT - 73T^{2}
79 1+0.630T+79T2 1 + 0.630T + 79T^{2}
83 1+(−0.711−0.711i)T+83iT2 1 + (-0.711 - 0.711i)T + 83iT^{2}
89 1−13.8iT−89T2 1 - 13.8iT - 89T^{2}
97 1−7.33T+97T2 1 - 7.33T + 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

−11.14955077456721127353967655464, −10.17000291884966990754959481169, −9.194756439242296117925974763129, −8.686956655309904202576388988109, −6.82129959031254348166754811545, −5.81807626669412653849735642693, −5.17898518698007257239821517490, −3.87557437151970181617503925616, −2.09662749487150721203025885434, −0.981301979300453239216381280897, 2.63362742949133578597468971440, 4.01550127516265924013384224275, 5.32462426008327453783382615952, 5.93089512161384195400221652244, 6.84616403698496361766682963791, 7.86219118722677301398056688245, 9.099821422539995033835233247054, 10.11286204138324785339682327476, 10.72964279463568956841571981009, 11.74914242063312613049080738423

Graph of the ZZ-function along the critical line