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Properties

Label 2-3800-152.37-c0-0-11
Degree 22
Conductor 38003800
Sign 0.923−0.382i0.923 - 0.382i
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  + (0.923 + 0.382i)2-s − 0.765·3-s + (0.707 + 0.707i)4-s + (−0.707 − 0.292i)6-s + (0.382 + 0.923i)8-s − 0.414·9-s − 1.41i·11-s + (−0.541 − 0.541i)12-s + 1.84·13-s + i·16-s + (−0.382 − 0.158i)18-s − i·19-s + (0.541 − 1.30i)22-s + (−0.292 − 0.707i)24-s + (1.70 + 0.707i)26-s + 1.08·27-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)2-s − 0.765·3-s + (0.707 + 0.707i)4-s + (−0.707 − 0.292i)6-s + (0.382 + 0.923i)8-s − 0.414·9-s − 1.41i·11-s + (−0.541 − 0.541i)12-s + 1.84·13-s + i·16-s + (−0.382 − 0.158i)18-s − i·19-s + (0.541 − 1.30i)22-s + (−0.292 − 0.707i)24-s + (1.70 + 0.707i)26-s + 1.08·27-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((0.923−0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s) L(s)=((0.923−0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: 0.923−0.382i0.923 - 0.382i
Analytic conductor: 1.896441.89644
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3800(1101,⋅)\chi_{3800} (1101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :0), 0.923−0.382i)(2,\ 3800,\ (\ :0),\ 0.923 - 0.382i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8518337561.851833756
L(12)L(\frac12) ≈\approx 1.8518337561.851833756
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+(−0.923−0.382i)T 1 + (-0.923 - 0.382i)T
5 1 1
19 1+iT 1 + iT
good3 1+0.765T+T2 1 + 0.765T + T^{2}
7 1+T2 1 + T^{2}
11 1+1.41iT−T2 1 + 1.41iT - T^{2}
13 1−1.84T+T2 1 - 1.84T + T^{2}
17 1+T2 1 + T^{2}
23 1+T2 1 + T^{2}
29 1+T2 1 + T^{2}
31 1−T2 1 - T^{2}
37 1−0.765T+T2 1 - 0.765T + T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1+T2 1 + T^{2}
53 1−0.765T+T2 1 - 0.765T + T^{2}
59 1+T2 1 + T^{2}
61 1−1.41iT−T2 1 - 1.41iT - T^{2}
67 1−1.84T+T2 1 - 1.84T + T^{2}
71 1−T2 1 - T^{2}
73 1+T2 1 + T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1−T2 1 - T^{2}
97 1−1.84iT−T2 1 - 1.84iT - 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

−8.475708389463429544358323057732, −8.065435389225415936691970083255, −6.83577293307994652504774740958, −6.32336242781281004614150650287, −5.74740494317913117737195227010, −5.21899487176826105519053680157, −4.16808109241270629287230093756, −3.41459005076754594451332044936, −2.64547138984168035875865359867, −1.05240540401779602092582256845, 1.21349285112509968177632613427, 2.16452128086605089260542088091, 3.34796517526064993247471554112, 4.06303022128499761815118834342, 4.88255402410686897452378287105, 5.62159947276026298022484212294, 6.25094567046066021880446425599, 6.76710714686817530044982694043, 7.77728386759656982788556856705, 8.592349117201555057430490976925

Graph of the ZZ-function along the critical line