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Properties

Label 2-312-104.11-c1-0-18
Degree 22
Conductor 312312
Sign 0.501+0.865i0.501 + 0.865i
Analytic cond. 2.491332.49133
Root an. cond. 1.578391.57839
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  + (−1.39 + 0.236i)2-s + (−0.5 − 0.866i)3-s + (1.88 − 0.659i)4-s + (1.86 − 1.86i)5-s + (0.901 + 1.08i)6-s + (4.71 − 1.26i)7-s + (−2.47 + 1.36i)8-s + (−0.499 + 0.866i)9-s + (−2.16 + 3.04i)10-s + (−1.99 − 0.535i)11-s + (−1.51 − 1.30i)12-s + (−3.15 + 1.75i)13-s + (−6.27 + 2.87i)14-s + (−2.54 − 0.683i)15-s + (3.13 − 2.48i)16-s + (3.84 + 2.22i)17-s + ⋯
L(s)  = 1  + (−0.985 + 0.167i)2-s + (−0.288 − 0.499i)3-s + (0.944 − 0.329i)4-s + (0.834 − 0.834i)5-s + (0.368 + 0.444i)6-s + (1.78 − 0.477i)7-s + (−0.875 + 0.482i)8-s + (−0.166 + 0.288i)9-s + (−0.683 + 0.962i)10-s + (−0.602 − 0.161i)11-s + (−0.437 − 0.376i)12-s + (−0.874 + 0.485i)13-s + (−1.67 + 0.768i)14-s + (−0.658 − 0.176i)15-s + (0.782 − 0.622i)16-s + (0.933 + 0.539i)17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.501+0.865i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+1/2) L(s)=((0.501+0.865i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.501+0.865i0.501 + 0.865i
Analytic conductor: 2.491332.49133
Root analytic conductor: 1.578391.57839
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ312(115,⋅)\chi_{312} (115, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 312, ( :1/2), 0.501+0.865i)(2,\ 312,\ (\ :1/2),\ 0.501 + 0.865i)

Particular Values

L(1)L(1) ≈\approx 0.873313−0.503040i0.873313 - 0.503040i
L(12)L(\frac12) ≈\approx 0.873313−0.503040i0.873313 - 0.503040i
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+(1.39−0.236i)T 1 + (1.39 - 0.236i)T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
13 1+(3.15−1.75i)T 1 + (3.15 - 1.75i)T
good5 1+(−1.86+1.86i)T−5iT2 1 + (-1.86 + 1.86i)T - 5iT^{2}
7 1+(−4.71+1.26i)T+(6.06−3.5i)T2 1 + (-4.71 + 1.26i)T + (6.06 - 3.5i)T^{2}
11 1+(1.99+0.535i)T+(9.52+5.5i)T2 1 + (1.99 + 0.535i)T + (9.52 + 5.5i)T^{2}
17 1+(−3.84−2.22i)T+(8.5+14.7i)T2 1 + (-3.84 - 2.22i)T + (8.5 + 14.7i)T^{2}
19 1+(−2.62+0.703i)T+(16.4−9.5i)T2 1 + (-2.62 + 0.703i)T + (16.4 - 9.5i)T^{2}
23 1+(3.41+5.91i)T+(−11.5+19.9i)T2 1 + (3.41 + 5.91i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.143−0.0829i)T+(14.5−25.1i)T2 1 + (0.143 - 0.0829i)T + (14.5 - 25.1i)T^{2}
31 1+(−3.77+3.77i)T−31iT2 1 + (-3.77 + 3.77i)T - 31iT^{2}
37 1+(1.19−4.46i)T+(−32.0−18.5i)T2 1 + (1.19 - 4.46i)T + (-32.0 - 18.5i)T^{2}
41 1+(2.06−7.71i)T+(−35.5−20.5i)T2 1 + (2.06 - 7.71i)T + (-35.5 - 20.5i)T^{2}
43 1+(7.03+4.05i)T+(21.5+37.2i)T2 1 + (7.03 + 4.05i)T + (21.5 + 37.2i)T^{2}
47 1+(1.87+1.87i)T+47iT2 1 + (1.87 + 1.87i)T + 47iT^{2}
53 1+8.22iT−53T2 1 + 8.22iT - 53T^{2}
59 1+(−0.352−1.31i)T+(−51.0+29.5i)T2 1 + (-0.352 - 1.31i)T + (-51.0 + 29.5i)T^{2}
61 1+(−0.601−0.347i)T+(30.5+52.8i)T2 1 + (-0.601 - 0.347i)T + (30.5 + 52.8i)T^{2}
67 1+(−1.26+4.71i)T+(−58.0−33.5i)T2 1 + (-1.26 + 4.71i)T + (-58.0 - 33.5i)T^{2}
71 1+(−2.46−9.19i)T+(−61.4+35.5i)T2 1 + (-2.46 - 9.19i)T + (-61.4 + 35.5i)T^{2}
73 1+(11.2−11.2i)T−73iT2 1 + (11.2 - 11.2i)T - 73iT^{2}
79 1−9.91iT−79T2 1 - 9.91iT - 79T^{2}
83 1+(−3.88−3.88i)T+83iT2 1 + (-3.88 - 3.88i)T + 83iT^{2}
89 1+(−13.1−3.52i)T+(77.0+44.5i)T2 1 + (-13.1 - 3.52i)T + (77.0 + 44.5i)T^{2}
97 1+(1.52−0.408i)T+(84.0−48.5i)T2 1 + (1.52 - 0.408i)T + (84.0 - 48.5i)T^{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

−11.51807197261609942934935617317, −10.41578858375802369427996569674, −9.734631765959103833956472587952, −8.319055664391217024072695677106, −8.051276313671524721127511649273, −6.87419066079891921933064138317, −5.56282761559221559158403153668, −4.84289030475277231853937895333, −2.17746095141961220483245340732, −1.17649219975015414106203239438, 1.81670692215019467775974361244, 3.00519428276545733137831523625, 5.07311411028045043374379905115, 5.83347494810950093993638082463, 7.37840425944829288059338266574, 7.966575243217084560687004749630, 9.225269165345992233676459435924, 10.14235759507906660518742642693, 10.60526045106149499654835448028, 11.66008086451382607732392078198

Graph of the ZZ-function along the critical line