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Properties

Label 2-1911-39.35-c0-0-5
Degree 22
Conductor 19111911
Sign −0.711+0.702i-0.711 + 0.702i
Analytic cond. 0.9537130.953713
Root an. cond. 0.9765820.976582
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.499 − 0.866i)9-s − 0.999·12-s + (0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + (−0.5 − 0.866i)19-s + 25-s − 0.999·27-s − 2·31-s + (−0.499 + 0.866i)36-s + (0.5 − 0.866i)37-s + (−0.499 − 0.866i)39-s + (0.5 + 0.866i)43-s + (0.499 + 0.866i)48-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.499 − 0.866i)9-s − 0.999·12-s + (0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + (−0.5 − 0.866i)19-s + 25-s − 0.999·27-s − 2·31-s + (−0.499 + 0.866i)36-s + (0.5 − 0.866i)37-s + (−0.499 − 0.866i)39-s + (0.5 + 0.866i)43-s + (0.499 + 0.866i)48-s + ⋯

Functional equation

Λ(s)=(1911s/2 ΓC(s) L(s)=((−0.711+0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1911s/2 ΓC(s) L(s)=((−0.711+0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19111911    =    3⋅72⋅133 \cdot 7^{2} \cdot 13
Sign: −0.711+0.702i-0.711 + 0.702i
Analytic conductor: 0.9537130.953713
Root analytic conductor: 0.9765820.976582
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1911(932,⋅)\chi_{1911} (932, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1911, ( :0), −0.711+0.702i)(2,\ 1911,\ (\ :0),\ -0.711 + 0.702i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0752775471.075277547
L(12)L(\frac12) ≈\approx 1.0752775471.075277547
L(1)L(1) 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)
bad3 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1 1
13 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good2 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
5 1−T2 1 - T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
31 1+2T+T2 1 + 2T + T^{2}
37 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
41 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
43 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1−T2 1 - T^{2}
53 1−T2 1 - T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
71 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
73 1−T+T2 1 - T + T^{2}
79 1−2T+T2 1 - 2T + T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)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

−9.055138029617111243271782938313, −8.433519947829370315942967901258, −7.54597049667449703388064285091, −6.72656206201205974021336890579, −5.94312688844674896336377359163, −5.23254816683026408419955155472, −4.11291657171887009703308756930, −3.04066019187052892660605350481, −1.93051786631205222761730056941, −0.76093550577132874140630587934, 2.02481444343192156444126202768, 3.21554307014154957171017323561, 3.87767303691356768798776058003, 4.57355223528254881244179533414, 5.46855858750128013691048401194, 6.61266398511722379631599483924, 7.61256818486104446635497022027, 8.251738121883670363180207639903, 9.104006363988677245194226858925, 9.281178927848861083330441581942

Graph of the ZZ-function along the critical line