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Properties

Label 2-3800-1.1-c1-0-7
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  − 0.848·3-s − 1.74·7-s − 2.28·9-s + 5.92·11-s − 6.78·13-s − 1.86·17-s − 19-s + 1.48·21-s − 5.94·23-s + 4.47·27-s + 3.29·29-s − 5.75·31-s − 5.02·33-s + 4.36·37-s + 5.75·39-s + 7.12·41-s + 6.98·43-s + 4.02·47-s − 3.95·49-s + 1.57·51-s + 9.19·53-s + 0.848·57-s + 2.51·59-s − 2.49·61-s + 3.97·63-s − 6.90·67-s + 5.04·69-s + ⋯
L(s)  = 1  − 0.489·3-s − 0.659·7-s − 0.760·9-s + 1.78·11-s − 1.88·13-s − 0.451·17-s − 0.229·19-s + 0.322·21-s − 1.23·23-s + 0.862·27-s + 0.612·29-s − 1.03·31-s − 0.874·33-s + 0.717·37-s + 0.921·39-s + 1.11·41-s + 1.06·43-s + 0.587·47-s − 0.565·49-s + 0.220·51-s + 1.26·53-s + 0.112·57-s + 0.327·59-s − 0.319·61-s + 0.501·63-s − 0.843·67-s + 0.606·69-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 0.95889209820.9588920982
L(12)L(\frac12) ≈\approx 0.95889209820.9588920982
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+0.848T+3T2 1 + 0.848T + 3T^{2}
7 1+1.74T+7T2 1 + 1.74T + 7T^{2}
11 1−5.92T+11T2 1 - 5.92T + 11T^{2}
13 1+6.78T+13T2 1 + 6.78T + 13T^{2}
17 1+1.86T+17T2 1 + 1.86T + 17T^{2}
23 1+5.94T+23T2 1 + 5.94T + 23T^{2}
29 1−3.29T+29T2 1 - 3.29T + 29T^{2}
31 1+5.75T+31T2 1 + 5.75T + 31T^{2}
37 1−4.36T+37T2 1 - 4.36T + 37T^{2}
41 1−7.12T+41T2 1 - 7.12T + 41T^{2}
43 1−6.98T+43T2 1 - 6.98T + 43T^{2}
47 1−4.02T+47T2 1 - 4.02T + 47T^{2}
53 1−9.19T+53T2 1 - 9.19T + 53T^{2}
59 1−2.51T+59T2 1 - 2.51T + 59T^{2}
61 1+2.49T+61T2 1 + 2.49T + 61T^{2}
67 1+6.90T+67T2 1 + 6.90T + 67T^{2}
71 1−1.27T+71T2 1 - 1.27T + 71T^{2}
73 1+12.1T+73T2 1 + 12.1T + 73T^{2}
79 1+13.8T+79T2 1 + 13.8T + 79T^{2}
83 1−4.94T+83T2 1 - 4.94T + 83T^{2}
89 1−15.6T+89T2 1 - 15.6T + 89T^{2}
97 1−15.7T+97T2 1 - 15.7T + 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

−8.684450401406013114131499073894, −7.58216666096238724875058967415, −6.97388978481289279810179985652, −6.17974203231934965052202009037, −5.74460954785962747724452208281, −4.61047569535495913284902192739, −4.02797270679690645626290552090, −2.92181657668714199430125623287, −2.05859302237289275087996841947, −0.55777705409897091411123674292, 0.55777705409897091411123674292, 2.05859302237289275087996841947, 2.92181657668714199430125623287, 4.02797270679690645626290552090, 4.61047569535495913284902192739, 5.74460954785962747724452208281, 6.17974203231934965052202009037, 6.97388978481289279810179985652, 7.58216666096238724875058967415, 8.684450401406013114131499073894

Graph of the ZZ-function along the critical line