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Properties

Label 2-3800-5.4-c1-0-25
Degree 22
Conductor 38003800
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 30.343130.3431
Root an. cond. 5.508465.50846
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.83i·3-s + 1.83i·7-s − 0.364·9-s + 0.834·11-s − 2.19i·13-s − 2.56i·17-s + 19-s − 3.36·21-s + 0.635i·23-s + 4.83i·27-s + 9.62·29-s − 6.59·31-s + 1.53i·33-s + 5.23i·37-s + 4.03·39-s + ⋯
L(s)  = 1  + 1.05i·3-s + 0.693i·7-s − 0.121·9-s + 0.251·11-s − 0.609i·13-s − 0.621i·17-s + 0.229·19-s − 0.734·21-s + 0.132i·23-s + 0.930i·27-s + 1.78·29-s − 1.18·31-s + 0.266i·33-s + 0.860i·37-s + 0.645·39-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s+1/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 30.343130.3431
Root analytic conductor: 5.508465.50846
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3800(3649,⋅)\chi_{3800} (3649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :1/2), −0.447−0.894i)(2,\ 3800,\ (\ :1/2),\ -0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.8619245361.861924536
L(12)L(\frac12) ≈\approx 1.8619245361.861924536
L(32)L(\frac{3}{2}) 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 1
5 1 1
19 1−T 1 - T
good3 1−1.83iT−3T2 1 - 1.83iT - 3T^{2}
7 1−1.83iT−7T2 1 - 1.83iT - 7T^{2}
11 1−0.834T+11T2 1 - 0.834T + 11T^{2}
13 1+2.19iT−13T2 1 + 2.19iT - 13T^{2}
17 1+2.56iT−17T2 1 + 2.56iT - 17T^{2}
23 1−0.635iT−23T2 1 - 0.635iT - 23T^{2}
29 1−9.62T+29T2 1 - 9.62T + 29T^{2}
31 1+6.59T+31T2 1 + 6.59T + 31T^{2}
37 1−5.23iT−37T2 1 - 5.23iT - 37T^{2}
41 1−4.43T+41T2 1 - 4.43T + 41T^{2}
43 1−7.06iT−43T2 1 - 7.06iT - 43T^{2}
47 1−9.86iT−47T2 1 - 9.86iT - 47T^{2}
53 1+0.668iT−53T2 1 + 0.668iT - 53T^{2}
59 1+0.397T+59T2 1 + 0.397T + 59T^{2}
61 1−2.26T+61T2 1 - 2.26T + 61T^{2}
67 1−2.43iT−67T2 1 - 2.43iT - 67T^{2}
71 1−4.12T+71T2 1 - 4.12T + 71T^{2}
73 1−9.49iT−73T2 1 - 9.49iT - 73T^{2}
79 1+2.62T+79T2 1 + 2.62T + 79T^{2}
83 1+10.8iT−83T2 1 + 10.8iT - 83T^{2}
89 1−5.97T+89T2 1 - 5.97T + 89T^{2}
97 1+10.5iT−97T2 1 + 10.5iT - 97T^{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

−8.908489495403596880836706590520, −8.131338473279670910477379655470, −7.31569831751891874675111182029, −6.41100390149538798418151000899, −5.59010999425794326914001971437, −4.91178129373782704811902515446, −4.25921993980672892949589834630, −3.27285231820678467931274711395, −2.60205291545495832822974439348, −1.16512632568380846914502097557, 0.61399921636274559247474804549, 1.58310955005510152360629385083, 2.40862489649484407571771670361, 3.69211282472426697328912494887, 4.30421526036297072065023350168, 5.36970284990587462464783735011, 6.29000741570215409440299144727, 6.90597384204016418864381360902, 7.35531945665832192930781275913, 8.153247589893984042804014398983

Graph of the ZZ-function along the critical line