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Properties

Label 2-3800-152.139-c0-0-6
Degree 22
Conductor 38003800
Sign −0.944+0.327i-0.944 + 0.327i
Analytic cond. 1.896441.89644
Root an. cond. 1.377111.37711
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  + (0.766 − 0.642i)2-s + (−0.326 − 0.118i)3-s + (0.173 − 0.984i)4-s + (−0.326 + 0.118i)6-s + (−0.500 − 0.866i)8-s + (−0.673 − 0.565i)9-s + (−0.173 − 0.300i)11-s + (−0.173 + 0.300i)12-s + (−0.939 − 0.342i)16-s + (1.53 − 1.28i)17-s − 0.879·18-s + (−0.939 + 0.342i)19-s + (−0.326 − 0.118i)22-s + (0.0603 + 0.342i)24-s + (0.326 + 0.565i)27-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.326 − 0.118i)3-s + (0.173 − 0.984i)4-s + (−0.326 + 0.118i)6-s + (−0.500 − 0.866i)8-s + (−0.673 − 0.565i)9-s + (−0.173 − 0.300i)11-s + (−0.173 + 0.300i)12-s + (−0.939 − 0.342i)16-s + (1.53 − 1.28i)17-s − 0.879·18-s + (−0.939 + 0.342i)19-s + (−0.326 − 0.118i)22-s + (0.0603 + 0.342i)24-s + (0.326 + 0.565i)27-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.944+0.327i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.944+0.327i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: −0.944+0.327i-0.944 + 0.327i
Analytic conductor: 1.896441.89644
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3800(1051,⋅)\chi_{3800} (1051, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :0), −0.944+0.327i)(2,\ 3800,\ (\ :0),\ -0.944 + 0.327i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3344754231.334475423
L(12)L(\frac12) ≈\approx 1.3344754231.334475423
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)
bad2 1+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
5 1 1
19 1+(0.939−0.342i)T 1 + (0.939 - 0.342i)T
good3 1+(0.326+0.118i)T+(0.766+0.642i)T2 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2}
7 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(0.173+0.300i)T+(−0.5+0.866i)T2 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2}
13 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
17 1+(−1.53+1.28i)T+(0.173−0.984i)T2 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2}
23 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
29 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−T2 1 - T^{2}
41 1+(1.43+0.524i)T+(0.766+0.642i)T2 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2}
43 1+(0.173+0.984i)T+(−0.939+0.342i)T2 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2}
47 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
53 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
59 1+(1.43−1.20i)T+(0.173−0.984i)T2 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2}
61 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
67 1+(−1.17−0.984i)T+(0.173+0.984i)T2 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2}
71 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
73 1+(1.43+0.524i)T+(0.766+0.642i)T2 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2}
79 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
83 1+(−0.939+1.62i)T+(−0.5−0.866i)T2 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2}
89 1+(−0.939+0.342i)T+(0.766−0.642i)T2 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2}
97 1+(−0.266+0.223i)T+(0.173−0.984i)T2 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)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

−8.504005184735619740129174072947, −7.42199575729730029518229857722, −6.68209315383449728255707631904, −5.85309521639287561963078924282, −5.40358468396883439418968532947, −4.57902630061509221961705527343, −3.47054503241654298779352632801, −3.03076114885464023737334460883, −1.85470866296079420774986080170, −0.58391819050418152457507952929, 1.84220947526851715491844553185, 2.93756536223408448999164366605, 3.70621111949113716651508636461, 4.71206751077087603506419840899, 5.20691098118497750988398324286, 6.11470862591100530793225752342, 6.45960219681886280507363385454, 7.63110860840224826233933024989, 8.069387725411025898371851100098, 8.668514962755325747850032143370

Graph of the ZZ-function along the critical line