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Properties

Label 2-1539-171.94-c0-0-0
Degree 22
Conductor 15391539
Sign −0.939−0.342i-0.939 - 0.342i
Analytic cond. 0.7680610.768061
Root an. cond. 0.8763900.876390
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)4-s + (−1 + 1.73i)5-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + (−0.499 − 0.866i)16-s − 17-s + 19-s + (−0.999 − 1.73i)20-s + (0.5 − 0.866i)23-s + (−1.49 − 2.59i)25-s − 0.999·28-s − 1.99·35-s + (0.5 + 0.866i)43-s − 0.999·44-s + (0.5 + 0.866i)47-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (−1 + 1.73i)5-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + (−0.499 − 0.866i)16-s − 17-s + 19-s + (−0.999 − 1.73i)20-s + (0.5 − 0.866i)23-s + (−1.49 − 2.59i)25-s − 0.999·28-s − 1.99·35-s + (0.5 + 0.866i)43-s − 0.999·44-s + (0.5 + 0.866i)47-s + ⋯

Functional equation

Λ(s)=(1539s/2 ΓC(s) L(s)=((−0.939−0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1539s/2 ΓC(s) L(s)=((−0.939−0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15391539    =    34⋅193^{4} \cdot 19
Sign: −0.939−0.342i-0.939 - 0.342i
Analytic conductor: 0.7680610.768061
Root analytic conductor: 0.8763900.876390
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1539(1405,⋅)\chi_{1539} (1405, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1539, ( :0), −0.939−0.342i)(2,\ 1539,\ (\ :0),\ -0.939 - 0.342i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.77856678660.7785667866
L(12)L(\frac12) ≈\approx 0.77856678660.7785667866
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 1
19 1−T 1 - T
good2 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
5 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
7 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
11 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
13 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+T+T2 1 + T + T^{2}
23 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−T2 1 - 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+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)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+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+T+T2 1 + T + T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.5−0.866i)T2 1 + (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

−9.991254375442355384779665473360, −9.058530300766132568082353512907, −8.346198557202789518640600511744, −7.46121563503960108225571751684, −7.08203191634842699397075795090, −6.14373512733645951645331919850, −4.72118727351530114599418890414, −4.07023875252508139076595206675, −3.04431156110291739140011353220, −2.38859209161679823867295914143, 0.69206603536665633115030562796, 1.47346561983698824087167778859, 3.69854526041394775218606109420, 4.28821936830943606031486906638, 5.07712747365096459048640149400, 5.67640108097000367348816951767, 7.01425878384551667760183793133, 7.84820460346061225194974113133, 8.710558031456455191830366852290, 9.054462350221225449902312093549

Graph of the ZZ-function along the critical line