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Properties

Label 2-1539-171.68-c0-0-0
Degree 22
Conductor 15391539
Sign 0.925−0.377i0.925 - 0.377i
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  + 4-s + (0.5 + 0.866i)7-s − 13-s + 16-s + 19-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)28-s + (0.5 − 0.866i)31-s − 37-s − 43-s − 52-s + (0.5 − 0.866i)61-s + 64-s − 67-s + (0.5 − 0.866i)73-s + ⋯
L(s)  = 1  + 4-s + (0.5 + 0.866i)7-s − 13-s + 16-s + 19-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)28-s + (0.5 − 0.866i)31-s − 37-s − 43-s − 52-s + (0.5 − 0.866i)61-s + 64-s − 67-s + (0.5 − 0.866i)73-s + ⋯

Functional equation

Λ(s)=(1539s/2 ΓC(s) L(s)=((0.925−0.377i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1539s/2 ΓC(s) L(s)=((0.925−0.377i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15391539    =    34⋅193^{4} \cdot 19
Sign: 0.925−0.377i0.925 - 0.377i
Analytic conductor: 0.7680610.768061
Root analytic conductor: 0.8763900.876390
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1539(296,⋅)\chi_{1539} (296, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1539, ( :0), 0.925−0.377i)(2,\ 1539,\ (\ :0),\ 0.925 - 0.377i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4362610261.436261026
L(12)L(\frac12) ≈\approx 1.4362610261.436261026
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−T2 1 - T^{2}
5 1+(0.5−0.866i)T2 1 + (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)T2 1 + (0.5 - 0.866i)T^{2}
13 1+T+T2 1 + T + T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1−T2 1 - T^{2}
29 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
31 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+T+T2 1 + T + T^{2}
41 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
43 1+T+T2 1 + T + T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)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+T+T2 1 + T + T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+T+T2 1 + T + T^{2}
83 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1−2T+T2 1 - 2T + 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

−9.756974307907275416324768914426, −8.911818576233452463272809483768, −7.906277925318936395148503235966, −7.39170794713973624650260959775, −6.47987503532712501661389572456, −5.55592948597905320287143075294, −4.96197525067685487367701666691, −3.50818336541310667620523576584, −2.55077014801167025377895925272, −1.68354271794708447845707629030, 1.33439010228231421956922834015, 2.49616759794252448577206942042, 3.50210846922613935971651846994, 4.63151513257975307605923674579, 5.47842805358497058530141016389, 6.54878749917281417990758012606, 7.25727411380110451526731399073, 7.75709023521877443322407063983, 8.676376175225585669733257482283, 9.957234692703789338578407344723

Graph of the ZZ-function along the critical line