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Properties

Label 2-3800-152.43-c0-0-4
Degree 22
Conductor 38003800
Sign −0.135+0.990i-0.135 + 0.990i
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  + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (1.32 − 0.766i)7-s + (0.866 + 0.500i)8-s + (0.939 + 0.342i)9-s + (0.939 − 1.62i)11-s + (−0.984 + 0.173i)13-s + (−1.17 − 0.984i)14-s + (0.173 − 0.984i)16-s − i·18-s + (0.939 − 0.342i)19-s + (−1.85 − 0.326i)22-s + (−0.223 − 0.266i)23-s + (0.5 + 0.866i)26-s + (−0.524 + 1.43i)28-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (1.32 − 0.766i)7-s + (0.866 + 0.500i)8-s + (0.939 + 0.342i)9-s + (0.939 − 1.62i)11-s + (−0.984 + 0.173i)13-s + (−1.17 − 0.984i)14-s + (0.173 − 0.984i)16-s − i·18-s + (0.939 − 0.342i)19-s + (−1.85 − 0.326i)22-s + (−0.223 − 0.266i)23-s + (0.5 + 0.866i)26-s + (−0.524 + 1.43i)28-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.135+0.990i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.135+0.990i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: −0.135+0.990i-0.135 + 0.990i
Analytic conductor: 1.896441.89644
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3800(651,⋅)\chi_{3800} (651, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :0), −0.135+0.990i)(2,\ 3800,\ (\ :0),\ -0.135 + 0.990i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3163588881.316358888
L(12)L(\frac12) ≈\approx 1.3163588881.316358888
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+(0.342+0.939i)T 1 + (0.342 + 0.939i)T
5 1 1
19 1+(−0.939+0.342i)T 1 + (-0.939 + 0.342i)T
good3 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
7 1+(−1.32+0.766i)T+(0.5−0.866i)T2 1 + (-1.32 + 0.766i)T + (0.5 - 0.866i)T^{2}
11 1+(−0.939+1.62i)T+(−0.5−0.866i)T2 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2}
13 1+(0.984−0.173i)T+(0.939−0.342i)T2 1 + (0.984 - 0.173i)T + (0.939 - 0.342i)T^{2}
17 1+(0.766−0.642i)T2 1 + (0.766 - 0.642i)T^{2}
23 1+(0.223+0.266i)T+(−0.173+0.984i)T2 1 + (0.223 + 0.266i)T + (-0.173 + 0.984i)T^{2}
29 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1−0.347iT−T2 1 - 0.347iT - T^{2}
41 1+(0.326−1.85i)T+(−0.939−0.342i)T2 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2}
43 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
47 1+(0.342−0.939i)T+(−0.766−0.642i)T2 1 + (0.342 - 0.939i)T + (-0.766 - 0.642i)T^{2}
53 1+(0.984+1.17i)T+(−0.173+0.984i)T2 1 + (0.984 + 1.17i)T + (-0.173 + 0.984i)T^{2}
59 1+(0.939−0.342i)T+(0.766−0.642i)T2 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2}
61 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
67 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
71 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
73 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
79 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
83 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
89 1+(0.0603+0.342i)T+(−0.939+0.342i)T2 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2}
97 1+(0.766−0.642i)T2 1 + (0.766 - 0.642i)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

−8.459424019596921319025735996074, −7.87551381704565072707940793792, −7.34185625306633107385137117118, −6.34433816117544612258188447220, −5.01078077269123625308352239872, −4.63804786731843626211659055779, −3.78629895476868934294499498766, −2.91951261642478425809591809181, −1.65871990825826427361377317461, −1.02978445708095806744248117483, 1.44429319361481778165190947211, 2.04905647114621949797222304320, 3.81961640945775764132732532941, 4.65504221496309316328198481292, 5.05202578907980099472286711061, 5.92465822453296900485343627978, 6.97518770134183356462937722738, 7.36049942651516457237368010889, 7.932739823396552466772707135940, 8.899817632867806001294316819252

Graph of the ZZ-function along the critical line