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Properties

Label 2-1920-24.11-c1-0-13
Degree 22
Conductor 19201920
Sign 0.764−0.644i0.764 - 0.644i
Analytic cond. 15.331215.3312
Root an. cond. 3.915513.91551
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  + (−1.72 − 0.146i)3-s − 5-s − 1.45i·7-s + (2.95 + 0.505i)9-s + 1.01i·11-s + 3.74i·13-s + (1.72 + 0.146i)15-s − 4.75i·17-s − 2.16·19-s + (−0.212 + 2.50i)21-s − 3.30·23-s + 25-s + (−5.02 − 1.30i)27-s + 3.59·29-s − 2.75i·31-s + ⋯
L(s)  = 1  + (−0.996 − 0.0845i)3-s − 0.447·5-s − 0.548i·7-s + (0.985 + 0.168i)9-s + 0.304i·11-s + 1.03i·13-s + (0.445 + 0.0378i)15-s − 1.15i·17-s − 0.497·19-s + (−0.0463 + 0.546i)21-s − 0.688·23-s + 0.200·25-s + (−0.967 − 0.251i)27-s + 0.667·29-s − 0.494i·31-s + ⋯

Functional equation

Λ(s)=(1920s/2 ΓC(s) L(s)=((0.764−0.644i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1920s/2 ΓC(s+1/2) L(s)=((0.764−0.644i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27⋅3⋅52^{7} \cdot 3 \cdot 5
Sign: 0.764−0.644i0.764 - 0.644i
Analytic conductor: 15.331215.3312
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1920(191,⋅)\chi_{1920} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1920, ( :1/2), 0.764−0.644i)(2,\ 1920,\ (\ :1/2),\ 0.764 - 0.644i)

Particular Values

L(1)L(1) ≈\approx 0.87687933710.8768793371
L(12)L(\frac12) ≈\approx 0.87687933710.8768793371
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
3 1+(1.72+0.146i)T 1 + (1.72 + 0.146i)T
5 1+T 1 + T
good7 1+1.45iT−7T2 1 + 1.45iT - 7T^{2}
11 1−1.01iT−11T2 1 - 1.01iT - 11T^{2}
13 1−3.74iT−13T2 1 - 3.74iT - 13T^{2}
17 1+4.75iT−17T2 1 + 4.75iT - 17T^{2}
19 1+2.16T+19T2 1 + 2.16T + 19T^{2}
23 1+3.30T+23T2 1 + 3.30T + 23T^{2}
29 1−3.59T+29T2 1 - 3.59T + 29T^{2}
31 1+2.75iT−31T2 1 + 2.75iT - 31T^{2}
37 1−6.06iT−37T2 1 - 6.06iT - 37T^{2}
41 1−7.32iT−41T2 1 - 7.32iT - 41T^{2}
43 1+0.0374T+43T2 1 + 0.0374T + 43T^{2}
47 1+5.62T+47T2 1 + 5.62T + 47T^{2}
53 1+4.31T+53T2 1 + 4.31T + 53T^{2}
59 1+11.9iT−59T2 1 + 11.9iT - 59T^{2}
61 1−5.30iT−61T2 1 - 5.30iT - 61T^{2}
67 1−8.03T+67T2 1 - 8.03T + 67T^{2}
71 1−12.4T+71T2 1 - 12.4T + 71T^{2}
73 1−10.3T+73T2 1 - 10.3T + 73T^{2}
79 1−11.3iT−79T2 1 - 11.3iT - 79T^{2}
83 1−11.1iT−83T2 1 - 11.1iT - 83T^{2}
89 1−6.31iT−89T2 1 - 6.31iT - 89T^{2}
97 1−9.48T+97T2 1 - 9.48T + 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

−9.625120471078868112492178040323, −8.361421122958981301545865184201, −7.59275626654194638851320188715, −6.75597741116184652687847477353, −6.37161980451479979261936717869, −5.06197977360274495132442149059, −4.53868363937306032404110546261, −3.68570043232716847730211354052, −2.19149991277073377018551821906, −0.870462147875926255756508244364, 0.50350524572257176345826306412, 1.97132099650509749959199006976, 3.37866997246513869966991488404, 4.22579450370266115154599743584, 5.22136573083537762454762280792, 5.89240149227160005637571689414, 6.54365499924677739877191881245, 7.57479355186190687122351555826, 8.289183050619146895635689944702, 9.059755785397804781880024615649

Graph of the ZZ-function along the critical line