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Properties

Label 2-368-16.13-c1-0-9
Degree 22
Conductor 368368
Sign −0.731−0.681i-0.731 - 0.681i
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  + (1.10 + 0.887i)2-s + (0.631 + 0.631i)3-s + (0.425 + 1.95i)4-s + (−2.74 + 2.74i)5-s + (0.135 + 1.25i)6-s − 1.52i·7-s + (−1.26 + 2.52i)8-s − 2.20i·9-s + (−5.44 + 0.585i)10-s + (−2.02 + 2.02i)11-s + (−0.965 + 1.50i)12-s + (2.71 + 2.71i)13-s + (1.35 − 1.68i)14-s − 3.46·15-s + (−3.63 + 1.66i)16-s + 3.66·17-s + ⋯
L(s)  = 1  + (0.778 + 0.627i)2-s + (0.364 + 0.364i)3-s + (0.212 + 0.977i)4-s + (−1.22 + 1.22i)5-s + (0.0551 + 0.512i)6-s − 0.577i·7-s + (−0.447 + 0.894i)8-s − 0.734i·9-s + (−1.72 + 0.185i)10-s + (−0.609 + 0.609i)11-s + (−0.278 + 0.433i)12-s + (0.753 + 0.753i)13-s + (0.362 − 0.450i)14-s − 0.893·15-s + (−0.909 + 0.415i)16-s + 0.888·17-s + ⋯

Functional equation

Λ(s)=(368s/2 ΓC(s) L(s)=((−0.731−0.681i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(368s/2 ΓC(s+1/2) L(s)=((−0.731−0.681i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 368368    =    24⋅232^{4} \cdot 23
Sign: −0.731−0.681i-0.731 - 0.681i
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), −0.731−0.681i)(2,\ 368,\ (\ :1/2),\ -0.731 - 0.681i)

Particular Values

L(1)L(1) ≈\approx 0.653864+1.66176i0.653864 + 1.66176i
L(12)L(\frac12) ≈\approx 0.653864+1.66176i0.653864 + 1.66176i
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+(−1.10−0.887i)T 1 + (-1.10 - 0.887i)T
23 1+iT 1 + iT
good3 1+(−0.631−0.631i)T+3iT2 1 + (-0.631 - 0.631i)T + 3iT^{2}
5 1+(2.74−2.74i)T−5iT2 1 + (2.74 - 2.74i)T - 5iT^{2}
7 1+1.52iT−7T2 1 + 1.52iT - 7T^{2}
11 1+(2.02−2.02i)T−11iT2 1 + (2.02 - 2.02i)T - 11iT^{2}
13 1+(−2.71−2.71i)T+13iT2 1 + (-2.71 - 2.71i)T + 13iT^{2}
17 1−3.66T+17T2 1 - 3.66T + 17T^{2}
19 1+(−1.20−1.20i)T+19iT2 1 + (-1.20 - 1.20i)T + 19iT^{2}
29 1+(−2.52−2.52i)T+29iT2 1 + (-2.52 - 2.52i)T + 29iT^{2}
31 1−2.74T+31T2 1 - 2.74T + 31T^{2}
37 1+(4.37−4.37i)T−37iT2 1 + (4.37 - 4.37i)T - 37iT^{2}
41 1−7.70iT−41T2 1 - 7.70iT - 41T^{2}
43 1+(−8.96+8.96i)T−43iT2 1 + (-8.96 + 8.96i)T - 43iT^{2}
47 1−4.24T+47T2 1 - 4.24T + 47T^{2}
53 1+(7.11−7.11i)T−53iT2 1 + (7.11 - 7.11i)T - 53iT^{2}
59 1+(−5.81+5.81i)T−59iT2 1 + (-5.81 + 5.81i)T - 59iT^{2}
61 1+(6.37+6.37i)T+61iT2 1 + (6.37 + 6.37i)T + 61iT^{2}
67 1+(0.0827+0.0827i)T+67iT2 1 + (0.0827 + 0.0827i)T + 67iT^{2}
71 1−6.62iT−71T2 1 - 6.62iT - 71T^{2}
73 1+7.91iT−73T2 1 + 7.91iT - 73T^{2}
79 1−6.42T+79T2 1 - 6.42T + 79T^{2}
83 1+(11.6+11.6i)T+83iT2 1 + (11.6 + 11.6i)T + 83iT^{2}
89 1+2.76iT−89T2 1 + 2.76iT - 89T^{2}
97 1+6.61T+97T2 1 + 6.61T + 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

−11.88836286112091698990610155580, −10.97794660765196669560705601413, −10.08121250753834210452631301938, −8.675452362437172014119323119940, −7.70569179779720876992325235466, −7.06160292293917870811064049054, −6.19435788348049679996988094711, −4.52854637059421615871476423970, −3.71695112095465242551714340924, −2.98487059449252050967082662796, 0.983335368265266889618073751129, 2.74559376709046905725743728081, 3.87637154399835357483845979744, 5.07898179216185446478130753850, 5.73419072072003425153551368408, 7.52322516862238707816996653425, 8.259384874211575978546265451813, 9.060522937713866997663256375865, 10.46536035536175299035096825456, 11.28959214732605859443085029322

Graph of the ZZ-function along the critical line