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Properties

Label 2-156-12.11-c1-0-18
Degree 22
Conductor 156156
Sign 0.995+0.0927i0.995 + 0.0927i
Analytic cond. 1.245661.24566
Root an. cond. 1.116091.11609
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  + (1.11 + 0.866i)2-s + (0.586 − 1.62i)3-s + (0.500 + 1.93i)4-s − 3.82i·5-s + (2.06 − 1.31i)6-s + 3.25i·7-s + (−1.11 + 2.59i)8-s + (−2.31 − 1.91i)9-s + (3.31 − 4.27i)10-s + (3.44 + 0.321i)12-s + 13-s + (−2.82 + 3.64i)14-s + (−6.23 − 2.24i)15-s + (−3.5 + 1.93i)16-s + 3.82i·17-s + (−0.928 − 4.13i)18-s + ⋯
L(s)  = 1  + (0.790 + 0.612i)2-s + (0.338 − 0.940i)3-s + (0.250 + 0.968i)4-s − 1.71i·5-s + (0.843 − 0.536i)6-s + 1.23i·7-s + (−0.395 + 0.918i)8-s + (−0.770 − 0.637i)9-s + (1.04 − 1.35i)10-s + (0.995 + 0.0927i)12-s + 0.277·13-s + (−0.754 + 0.973i)14-s + (−1.60 − 0.579i)15-s + (−0.875 + 0.484i)16-s + 0.927i·17-s + (−0.218 − 0.975i)18-s + ⋯

Functional equation

Λ(s)=(156s/2 ΓC(s) L(s)=((0.995+0.0927i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0927i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(156s/2 ΓC(s+1/2) L(s)=((0.995+0.0927i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0927i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 156156    =    22⋅3⋅132^{2} \cdot 3 \cdot 13
Sign: 0.995+0.0927i0.995 + 0.0927i
Analytic conductor: 1.245661.24566
Root analytic conductor: 1.116091.11609
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ156(131,⋅)\chi_{156} (131, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 156, ( :1/2), 0.995+0.0927i)(2,\ 156,\ (\ :1/2),\ 0.995 + 0.0927i)

Particular Values

L(1)L(1) ≈\approx 1.75820−0.0816774i1.75820 - 0.0816774i
L(12)L(\frac12) ≈\approx 1.75820−0.0816774i1.75820 - 0.0816774i
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+(−1.11−0.866i)T 1 + (-1.11 - 0.866i)T
3 1+(−0.586+1.62i)T 1 + (-0.586 + 1.62i)T
13 1−T 1 - T
good5 1+3.82iT−5T2 1 + 3.82iT - 5T^{2}
7 1−3.25iT−7T2 1 - 3.25iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
17 1−3.82iT−17T2 1 - 3.82iT - 17T^{2}
19 1−5.29iT−19T2 1 - 5.29iT - 19T^{2}
23 1+2.34T+23T2 1 + 2.34T + 23T^{2}
29 1+6.92iT−29T2 1 + 6.92iT - 29T^{2}
31 1+5.29iT−31T2 1 + 5.29iT - 31T^{2}
37 1−4.62T+37T2 1 - 4.62T + 37T^{2}
41 1−0.719iT−41T2 1 - 0.719iT - 41T^{2}
43 1+7.32iT−43T2 1 + 7.32iT - 43T^{2}
47 1+1.17T+47T2 1 + 1.17T + 47T^{2}
53 1+0.719iT−53T2 1 + 0.719iT - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1−5.29iT−67T2 1 - 5.29iT - 67T^{2}
71 1+1.17T+71T2 1 + 1.17T + 71T^{2}
73 1−7.24T+73T2 1 - 7.24T + 73T^{2}
79 1+1.22iT−79T2 1 + 1.22iT - 79T^{2}
83 1−4.69T+83T2 1 - 4.69T + 83T^{2}
89 1−14.5iT−89T2 1 - 14.5iT - 89T^{2}
97 1−15.2T+97T2 1 - 15.2T + 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

−12.91356696851178288505864338183, −12.26061510270211812766745835562, −11.73329766225640396738675509262, −9.333414580402539154643540154145, −8.369000663485634391513115533095, −7.964498364205499229963460785611, −6.10669187462974166608575654638, −5.56847037861222249784840693986, −4.01421201245683101786059422187, −2.05868639853871954857846462661, 2.77878981890764554380270789862, 3.63137557525668226292285310873, 4.84388369322217865405670124977, 6.47253845551160775631889560701, 7.40565288617832594864250603661, 9.366696825054488723033336265198, 10.36742057514727639479689525036, 10.84187856177014997023355865548, 11.54236834620042240545290466360, 13.32258018615220907837873379814

Graph of the ZZ-function along the critical line