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Properties

Label 2-3800-1.1-c1-0-63
Degree 22
Conductor 38003800
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.664·3-s + 0.345·7-s − 2.55·9-s + 3.51·11-s + 3.22·13-s − 4.88·17-s + 19-s − 0.229·21-s − 7.62·23-s + 3.69·27-s − 1.73·29-s − 4.76·31-s − 2.33·33-s + 1.86·37-s − 2.14·39-s + 6.76·41-s − 0.606·43-s − 3.34·47-s − 6.88·49-s + 3.24·51-s + 0.107·53-s − 0.664·57-s + 12.3·59-s − 8.41·61-s − 0.884·63-s − 2.23·67-s + 5.06·69-s + ⋯
L(s)  = 1  − 0.383·3-s + 0.130·7-s − 0.852·9-s + 1.06·11-s + 0.894·13-s − 1.18·17-s + 0.229·19-s − 0.0501·21-s − 1.59·23-s + 0.710·27-s − 0.323·29-s − 0.855·31-s − 0.406·33-s + 0.307·37-s − 0.343·39-s + 1.05·41-s − 0.0924·43-s − 0.488·47-s − 0.982·49-s + 0.454·51-s + 0.0147·53-s − 0.0880·57-s + 1.60·59-s − 1.07·61-s − 0.111·63-s − 0.273·67-s + 0.610·69-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: −1-1
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: 11
Selberg data: (2, 3800, ( :1/2), −1)(2,\ 3800,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+0.664T+3T2 1 + 0.664T + 3T^{2}
7 1−0.345T+7T2 1 - 0.345T + 7T^{2}
11 1−3.51T+11T2 1 - 3.51T + 11T^{2}
13 1−3.22T+13T2 1 - 3.22T + 13T^{2}
17 1+4.88T+17T2 1 + 4.88T + 17T^{2}
23 1+7.62T+23T2 1 + 7.62T + 23T^{2}
29 1+1.73T+29T2 1 + 1.73T + 29T^{2}
31 1+4.76T+31T2 1 + 4.76T + 31T^{2}
37 1−1.86T+37T2 1 - 1.86T + 37T^{2}
41 1−6.76T+41T2 1 - 6.76T + 41T^{2}
43 1+0.606T+43T2 1 + 0.606T + 43T^{2}
47 1+3.34T+47T2 1 + 3.34T + 47T^{2}
53 1−0.107T+53T2 1 - 0.107T + 53T^{2}
59 1−12.3T+59T2 1 - 12.3T + 59T^{2}
61 1+8.41T+61T2 1 + 8.41T + 61T^{2}
67 1+2.23T+67T2 1 + 2.23T + 67T^{2}
71 1+0.536T+71T2 1 + 0.536T + 71T^{2}
73 1+5.57T+73T2 1 + 5.57T + 73T^{2}
79 1+2.67T+79T2 1 + 2.67T + 79T^{2}
83 1+1.79T+83T2 1 + 1.79T + 83T^{2}
89 1−8.94T+89T2 1 - 8.94T + 89T^{2}
97 1+10.1T+97T2 1 + 10.1T + 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

−8.245098151402001525541520375792, −7.38258184101036106601574655264, −6.32538285869862024986811479092, −6.14835844746175876651844696368, −5.19472068512857797717119603548, −4.20519312046796208485193517784, −3.60768974297536016359389490388, −2.42444030761923594410837368434, −1.40249609117630693939587679186, 0, 1.40249609117630693939587679186, 2.42444030761923594410837368434, 3.60768974297536016359389490388, 4.20519312046796208485193517784, 5.19472068512857797717119603548, 6.14835844746175876651844696368, 6.32538285869862024986811479092, 7.38258184101036106601574655264, 8.245098151402001525541520375792

Graph of the ZZ-function along the critical line