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Properties

Label 2-3800-152.131-c0-0-1
Degree 22
Conductor 38003800
Sign 0.0158−0.999i0.0158 - 0.999i
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  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (−1.62 + 0.939i)7-s + (0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (−0.173 + 0.300i)11-s + (0.642 + 0.766i)13-s + (−1.76 + 0.642i)14-s + (0.766 + 0.642i)16-s − 0.999i·18-s + (−0.173 + 0.984i)19-s + (−0.223 + 0.266i)22-s + (−0.524 + 1.43i)23-s + (0.5 + 0.866i)26-s + (−1.85 + 0.326i)28-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (−1.62 + 0.939i)7-s + (0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (−0.173 + 0.300i)11-s + (0.642 + 0.766i)13-s + (−1.76 + 0.642i)14-s + (0.766 + 0.642i)16-s − 0.999i·18-s + (−0.173 + 0.984i)19-s + (−0.223 + 0.266i)22-s + (−0.524 + 1.43i)23-s + (0.5 + 0.866i)26-s + (−1.85 + 0.326i)28-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((0.0158−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0158 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s) L(s)=((0.0158−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0158 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: 0.0158−0.999i0.0158 - 0.999i
Analytic conductor: 1.896441.89644
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3800(1651,⋅)\chi_{3800} (1651, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :0), 0.0158−0.999i)(2,\ 3800,\ (\ :0),\ 0.0158 - 0.999i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8964707271.896470727
L(12)L(\frac12) ≈\approx 1.8964707271.896470727
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+(−0.984−0.173i)T 1 + (-0.984 - 0.173i)T
5 1 1
19 1+(0.173−0.984i)T 1 + (0.173 - 0.984i)T
good3 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
7 1+(1.62−0.939i)T+(0.5−0.866i)T2 1 + (1.62 - 0.939i)T + (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+(−0.642−0.766i)T+(−0.173+0.984i)T2 1 + (-0.642 - 0.766i)T + (-0.173 + 0.984i)T^{2}
17 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
23 1+(0.524−1.43i)T+(−0.766−0.642i)T2 1 + (0.524 - 1.43i)T + (-0.766 - 0.642i)T^{2}
29 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1−1.53iT−T2 1 - 1.53iT - T^{2}
41 1+(−0.266−0.223i)T+(0.173+0.984i)T2 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2}
43 1+(0.766−0.642i)T2 1 + (0.766 - 0.642i)T^{2}
47 1+(−0.984+0.173i)T+(0.939−0.342i)T2 1 + (-0.984 + 0.173i)T + (0.939 - 0.342i)T^{2}
53 1+(−0.642+1.76i)T+(−0.766−0.642i)T2 1 + (-0.642 + 1.76i)T + (-0.766 - 0.642i)T^{2}
59 1+(−0.173+0.984i)T+(−0.939−0.342i)T2 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2}
61 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
67 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
71 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
73 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
79 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
83 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
89 1+(1.17−0.984i)T+(0.173−0.984i)T2 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2}
97 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)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

−8.835804266530214422716900350144, −8.092098597735882409760399676329, −6.96546758081173651114206750943, −6.50524459964501095070624854249, −5.91800571150585852249597302897, −5.34841066461703795745045914192, −4.00480182367781979592688722360, −3.55598764224215274574476419270, −2.80366746769176514471843634747, −1.71637813427509189513980870520, 0.74751150436094506008877153765, 2.44514766464527012564157558600, 2.97341384824334508680579613313, 3.94143699379803011195751617190, 4.48901229309227030261335219642, 5.67897811625941526547607156317, 6.04239541180613775939514651349, 6.99351047441964366713600417438, 7.41910970068768186657221465644, 8.406013416656313386893228889050

Graph of the ZZ-function along the critical line