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Properties

Label 2-3800-5.4-c1-0-20
Degree 22
Conductor 38003800
Sign −0.894+0.447i-0.894 + 0.447i
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  + 1.76i·3-s + 4.62i·7-s − 0.103·9-s − 5.52·11-s + 5.49i·13-s + 6.62i·17-s + 19-s − 8.14·21-s + 4.14i·23-s + 5.10i·27-s + 7.87·29-s + 1.25·31-s − 9.72i·33-s + 0.387i·37-s − 9.67·39-s + ⋯
L(s)  = 1  + 1.01i·3-s + 1.74i·7-s − 0.0343·9-s − 1.66·11-s + 1.52i·13-s + 1.60i·17-s + 0.229·19-s − 1.77·21-s + 0.865i·23-s + 0.982i·27-s + 1.46·29-s + 0.224·31-s − 1.69i·33-s + 0.0637i·37-s − 1.54·39-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: −0.894+0.447i-0.894 + 0.447i
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), −0.894+0.447i)(2,\ 3800,\ (\ :1/2),\ -0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.5305563711.530556371
L(12)L(\frac12) ≈\approx 1.5305563711.530556371
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 1−1.76iT−3T2 1 - 1.76iT - 3T^{2}
7 1−4.62iT−7T2 1 - 4.62iT - 7T^{2}
11 1+5.52T+11T2 1 + 5.52T + 11T^{2}
13 1−5.49iT−13T2 1 - 5.49iT - 13T^{2}
17 1−6.62iT−17T2 1 - 6.62iT - 17T^{2}
23 1−4.14iT−23T2 1 - 4.14iT - 23T^{2}
29 1−7.87T+29T2 1 - 7.87T + 29T^{2}
31 1−1.25T+31T2 1 - 1.25T + 31T^{2}
37 1−0.387iT−37T2 1 - 0.387iT - 37T^{2}
41 1−6.77T+41T2 1 - 6.77T + 41T^{2}
43 1+10.9iT−43T2 1 + 10.9iT - 43T^{2}
47 1+1.72iT−47T2 1 + 1.72iT - 47T^{2}
53 1−1.49iT−53T2 1 - 1.49iT - 53T^{2}
59 1+0.626T+59T2 1 + 0.626T + 59T^{2}
61 1−15.0T+61T2 1 - 15.0T + 61T^{2}
67 1+5.22iT−67T2 1 + 5.22iT - 67T^{2}
71 1+11.0T+71T2 1 + 11.0T + 71T^{2}
73 1+4.83iT−73T2 1 + 4.83iT - 73T^{2}
79 1−2.98T+79T2 1 - 2.98T + 79T^{2}
83 1+2.74iT−83T2 1 + 2.74iT - 83T^{2}
89 1+4.27T+89T2 1 + 4.27T + 89T^{2}
97 1−13.7iT−97T2 1 - 13.7iT - 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

−8.870331992407754769187663822878, −8.479531534791128939491059187719, −7.58999428143204628278591229663, −6.53895679688382680505264045220, −5.73310335969911330593803927912, −5.18080520841151886140763018977, −4.45010469299731146659966239112, −3.55775146505649939942628148591, −2.55965775432027961233084766395, −1.82820213190766681649474878966, 0.54236866150373580835468853725, 0.949178259634359153352993798689, 2.56754261758885739691919033778, 3.05364215897853056653497623342, 4.40923591793889645791041773897, 4.97342780122875455047696196600, 5.95484426972351864196793547063, 6.91259796970260692414781318447, 7.36718153166328054233246870416, 7.889893680909231456890231369774

Graph of the ZZ-function along the critical line