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Properties

Label 2-1539-171.140-c0-0-0
Degree 22
Conductor 15391539
Sign 0.999+0.0383i0.999 + 0.0383i
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 + (0.5 + 0.866i)7-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + 19-s + 25-s + (0.499 − 0.866i)28-s + (0.5 − 0.866i)31-s − 37-s + (0.5 − 0.866i)43-s + (0.499 − 0.866i)52-s − 61-s + 0.999·64-s + (0.5 + 0.866i)67-s + (0.5 + 0.866i)73-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + 19-s + 25-s + (0.499 − 0.866i)28-s + (0.5 − 0.866i)31-s − 37-s + (0.5 − 0.866i)43-s + (0.499 − 0.866i)52-s − 61-s + 0.999·64-s + (0.5 + 0.866i)67-s + (0.5 + 0.866i)73-s + ⋯

Functional equation

Λ(s)=(1539s/2 ΓC(s) L(s)=((0.999+0.0383i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0383i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1539s/2 ΓC(s) L(s)=((0.999+0.0383i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0383i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15391539    =    34⋅193^{4} \cdot 19
Sign: 0.999+0.0383i0.999 + 0.0383i
Analytic conductor: 0.7680610.768061
Root analytic conductor: 0.8763900.876390
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1539(539,⋅)\chi_{1539} (539, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1539, ( :0), 0.999+0.0383i)(2,\ 1539,\ (\ :0),\ 0.999 + 0.0383i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0809242211.080924221
L(12)L(\frac12) ≈\approx 1.0809242211.080924221
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−T2 1 - 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+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1−T2 1 - 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−T2 1 - 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+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1−T2 1 - T^{2}
61 1+T+T2 1 + T + T^{2}
67 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)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+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)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+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)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

−9.498820869443401071869200797126, −8.930049294431894290673833486777, −8.312841096688338056052376035286, −7.14780447582695294703046623662, −6.25167544764299354818661998096, −5.45884459273746572262406937792, −4.82462551101106316752878679549, −3.80936670065825329478914242462, −2.40440856173645691182839201906, −1.30346006937395725879944143999, 1.12656888326601167268129999208, 2.91169442020472748113310662599, 3.62598008450832854572115211491, 4.61906893374022407338417441227, 5.30639795202839473766471746115, 6.59428820174605517888348280638, 7.45854974343366795451643102798, 8.001306040419517469551993195392, 8.740307741070049091456945951617, 9.561817816199146173121503217129

Graph of the ZZ-function along the critical line