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Properties

Label 2-3800-1.1-c1-0-19
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  − 1.93·3-s + 1.24·7-s + 0.747·9-s − 0.513·11-s + 6.15·13-s − 4.51·17-s − 19-s − 2.41·21-s − 5.86·23-s + 4.36·27-s + 6.62·29-s + 6.41·31-s + 0.994·33-s − 1.40·37-s − 11.9·39-s + 10.6·41-s − 3.04·43-s − 1.99·47-s − 5.44·49-s + 8.74·51-s − 14.0·53-s + 1.93·57-s − 4.34·59-s + 10.7·61-s + 0.932·63-s − 9.89·67-s + 11.3·69-s + ⋯
L(s)  = 1  − 1.11·3-s + 0.471·7-s + 0.249·9-s − 0.154·11-s + 1.70·13-s − 1.09·17-s − 0.229·19-s − 0.526·21-s − 1.22·23-s + 0.839·27-s + 1.23·29-s + 1.15·31-s + 0.173·33-s − 0.231·37-s − 1.90·39-s + 1.66·41-s − 0.464·43-s − 0.290·47-s − 0.777·49-s + 1.22·51-s − 1.93·53-s + 0.256·57-s − 0.565·59-s + 1.37·61-s + 0.117·63-s − 1.20·67-s + 1.36·69-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 1.1878567091.187856709
L(12)L(\frac12) ≈\approx 1.1878567091.187856709
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+1.93T+3T2 1 + 1.93T + 3T^{2}
7 1−1.24T+7T2 1 - 1.24T + 7T^{2}
11 1+0.513T+11T2 1 + 0.513T + 11T^{2}
13 1−6.15T+13T2 1 - 6.15T + 13T^{2}
17 1+4.51T+17T2 1 + 4.51T + 17T^{2}
23 1+5.86T+23T2 1 + 5.86T + 23T^{2}
29 1−6.62T+29T2 1 - 6.62T + 29T^{2}
31 1−6.41T+31T2 1 - 6.41T + 31T^{2}
37 1+1.40T+37T2 1 + 1.40T + 37T^{2}
41 1−10.6T+41T2 1 - 10.6T + 41T^{2}
43 1+3.04T+43T2 1 + 3.04T + 43T^{2}
47 1+1.99T+47T2 1 + 1.99T + 47T^{2}
53 1+14.0T+53T2 1 + 14.0T + 53T^{2}
59 1+4.34T+59T2 1 + 4.34T + 59T^{2}
61 1−10.7T+61T2 1 - 10.7T + 61T^{2}
67 1+9.89T+67T2 1 + 9.89T + 67T^{2}
71 1−7.42T+71T2 1 - 7.42T + 71T^{2}
73 1−12.8T+73T2 1 - 12.8T + 73T^{2}
79 1−2.56T+79T2 1 - 2.56T + 79T^{2}
83 1+7.50T+83T2 1 + 7.50T + 83T^{2}
89 1+7.85T+89T2 1 + 7.85T + 89T^{2}
97 1−6.74T+97T2 1 - 6.74T + 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

−8.329969858292361094229887419795, −7.983451765466953643050475616939, −6.50284192652144108068409233795, −6.44479818527058019354710437964, −5.61983152234551653815331344414, −4.71292219796688939119103390979, −4.15706055029957661762160157046, −2.98833209566256860990705508822, −1.77939889325735359663450105121, −0.68198939034702020857083765837, 0.68198939034702020857083765837, 1.77939889325735359663450105121, 2.98833209566256860990705508822, 4.15706055029957661762160157046, 4.71292219796688939119103390979, 5.61983152234551653815331344414, 6.44479818527058019354710437964, 6.50284192652144108068409233795, 7.983451765466953643050475616939, 8.329969858292361094229887419795

Graph of the ZZ-function along the critical line