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Properties

Label 2-3800-1.1-c1-0-4
Degree 22
Conductor 38003800
Sign 11
Analytic cond. 30.343130.3431
Root an. cond. 5.508465.50846
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.30·3-s + 1.63·7-s + 7.89·9-s − 4.67·11-s − 4.75·13-s − 1.41·17-s + 19-s − 5.38·21-s − 1.96·23-s − 16.1·27-s + 6.85·29-s + 5.08·31-s + 15.4·33-s + 10.6·37-s + 15.6·39-s − 4.52·41-s − 7.83·43-s − 10.9·47-s − 4.34·49-s + 4.66·51-s − 1.55·53-s − 3.30·57-s − 6.81·59-s − 0.109·61-s + 12.8·63-s − 10.5·67-s + 6.48·69-s + ⋯
L(s)  = 1  − 1.90·3-s + 0.616·7-s + 2.63·9-s − 1.40·11-s − 1.31·13-s − 0.343·17-s + 0.229·19-s − 1.17·21-s − 0.409·23-s − 3.11·27-s + 1.27·29-s + 0.912·31-s + 2.68·33-s + 1.74·37-s + 2.51·39-s − 0.706·41-s − 1.19·43-s − 1.59·47-s − 0.620·49-s + 0.653·51-s − 0.214·53-s − 0.437·57-s − 0.887·59-s − 0.0139·61-s + 1.62·63-s − 1.29·67-s + 0.780·69-s + ⋯

Functional equation

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

Invariants

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

Particular Values

L(1)L(1) ≈\approx 0.56291889110.5629188911
L(12)L(\frac12) ≈\approx 0.56291889110.5629188911
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+3.30T+3T2 1 + 3.30T + 3T^{2}
7 1−1.63T+7T2 1 - 1.63T + 7T^{2}
11 1+4.67T+11T2 1 + 4.67T + 11T^{2}
13 1+4.75T+13T2 1 + 4.75T + 13T^{2}
17 1+1.41T+17T2 1 + 1.41T + 17T^{2}
23 1+1.96T+23T2 1 + 1.96T + 23T^{2}
29 1−6.85T+29T2 1 - 6.85T + 29T^{2}
31 1−5.08T+31T2 1 - 5.08T + 31T^{2}
37 1−10.6T+37T2 1 - 10.6T + 37T^{2}
41 1+4.52T+41T2 1 + 4.52T + 41T^{2}
43 1+7.83T+43T2 1 + 7.83T + 43T^{2}
47 1+10.9T+47T2 1 + 10.9T + 47T^{2}
53 1+1.55T+53T2 1 + 1.55T + 53T^{2}
59 1+6.81T+59T2 1 + 6.81T + 59T^{2}
61 1+0.109T+61T2 1 + 0.109T + 61T^{2}
67 1+10.5T+67T2 1 + 10.5T + 67T^{2}
71 1−12.7T+71T2 1 - 12.7T + 71T^{2}
73 1−0.519T+73T2 1 - 0.519T + 73T^{2}
79 1−0.840T+79T2 1 - 0.840T + 79T^{2}
83 1−11.9T+83T2 1 - 11.9T + 83T^{2}
89 1+7.44T+89T2 1 + 7.44T + 89T^{2}
97 1−8.85T+97T2 1 - 8.85T + 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

−8.120624589273801885028025419296, −7.76468598900464449070858832962, −6.80497102187186364818282763424, −6.27765077601011522401563841547, −5.32764072181234475506334312861, −4.85837984769410735562740723784, −4.46648786347518582615362076038, −2.86946963991985976446731019380, −1.74476405464845389362770201299, −0.47486822745821865085859932953, 0.47486822745821865085859932953, 1.74476405464845389362770201299, 2.86946963991985976446731019380, 4.46648786347518582615362076038, 4.85837984769410735562740723784, 5.32764072181234475506334312861, 6.27765077601011522401563841547, 6.80497102187186364818282763424, 7.76468598900464449070858832962, 8.120624589273801885028025419296

Graph of the ZZ-function along the critical line