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Properties

Label 2-3800-1.1-c1-0-37
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  − 2.43·3-s − 3.60·7-s + 2.90·9-s − 2.37·11-s + 3.96·13-s − 4.28·17-s + 19-s + 8.76·21-s + 1.07·23-s + 0.227·27-s + 9.47·29-s + 6.38·31-s + 5.78·33-s − 2.04·37-s − 9.63·39-s − 4.38·41-s − 7.86·43-s − 3.83·47-s + 6.01·49-s + 10.4·51-s + 11.7·53-s − 2.43·57-s + 4.59·59-s + 6.62·61-s − 10.4·63-s − 7.02·67-s − 2.60·69-s + ⋯
L(s)  = 1  − 1.40·3-s − 1.36·7-s + 0.968·9-s − 0.717·11-s + 1.10·13-s − 1.03·17-s + 0.229·19-s + 1.91·21-s + 0.223·23-s + 0.0437·27-s + 1.75·29-s + 1.14·31-s + 1.00·33-s − 0.336·37-s − 1.54·39-s − 0.684·41-s − 1.19·43-s − 0.559·47-s + 0.859·49-s + 1.45·51-s + 1.61·53-s − 0.321·57-s + 0.598·59-s + 0.848·61-s − 1.32·63-s − 0.858·67-s − 0.314·69-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+2.43T+3T2 1 + 2.43T + 3T^{2}
7 1+3.60T+7T2 1 + 3.60T + 7T^{2}
11 1+2.37T+11T2 1 + 2.37T + 11T^{2}
13 1−3.96T+13T2 1 - 3.96T + 13T^{2}
17 1+4.28T+17T2 1 + 4.28T + 17T^{2}
23 1−1.07T+23T2 1 - 1.07T + 23T^{2}
29 1−9.47T+29T2 1 - 9.47T + 29T^{2}
31 1−6.38T+31T2 1 - 6.38T + 31T^{2}
37 1+2.04T+37T2 1 + 2.04T + 37T^{2}
41 1+4.38T+41T2 1 + 4.38T + 41T^{2}
43 1+7.86T+43T2 1 + 7.86T + 43T^{2}
47 1+3.83T+47T2 1 + 3.83T + 47T^{2}
53 1−11.7T+53T2 1 - 11.7T + 53T^{2}
59 1−4.59T+59T2 1 - 4.59T + 59T^{2}
61 1−6.62T+61T2 1 - 6.62T + 61T^{2}
67 1+7.02T+67T2 1 + 7.02T + 67T^{2}
71 1−4.99T+71T2 1 - 4.99T + 71T^{2}
73 1−2.93T+73T2 1 - 2.93T + 73T^{2}
79 1−0.860T+79T2 1 - 0.860T + 79T^{2}
83 1+11.9T+83T2 1 + 11.9T + 83T^{2}
89 1−13.6T+89T2 1 - 13.6T + 89T^{2}
97 1+18.5T+97T2 1 + 18.5T + 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

−8.269804930161372473796696041868, −6.87214967880358875786595777107, −6.65194225880039696802627093771, −5.99990796320943991407626417545, −5.23459269534444572600849741679, −4.48776180496958968650406602687, −3.46172131490042659473916346950, −2.58811194199752644047495856550, −1.02054784796983291449167355497, 0, 1.02054784796983291449167355497, 2.58811194199752644047495856550, 3.46172131490042659473916346950, 4.48776180496958968650406602687, 5.23459269534444572600849741679, 5.99990796320943991407626417545, 6.65194225880039696802627093771, 6.87214967880358875786595777107, 8.269804930161372473796696041868

Graph of the ZZ-function along the critical line