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Properties

Label 2-3800-1.1-c1-0-3
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  − 0.486·3-s − 3.63·7-s − 2.76·9-s − 2.79·11-s − 2.86·13-s − 1.17·17-s + 19-s + 1.76·21-s + 0.617·23-s + 2.80·27-s − 4.96·29-s + 0.745·31-s + 1.36·33-s − 8.23·37-s + 1.39·39-s + 9.98·41-s − 10.4·43-s − 5.07·47-s + 6.19·49-s + 0.571·51-s + 7.45·53-s − 0.486·57-s + 3.83·59-s + 11.2·61-s + 10.0·63-s − 6.10·67-s − 0.300·69-s + ⋯
L(s)  = 1  − 0.280·3-s − 1.37·7-s − 0.921·9-s − 0.842·11-s − 0.794·13-s − 0.284·17-s + 0.229·19-s + 0.385·21-s + 0.128·23-s + 0.539·27-s − 0.922·29-s + 0.133·31-s + 0.236·33-s − 1.35·37-s + 0.223·39-s + 1.55·41-s − 1.60·43-s − 0.740·47-s + 0.885·49-s + 0.0800·51-s + 1.02·53-s − 0.0644·57-s + 0.499·59-s + 1.43·61-s + 1.26·63-s − 0.746·67-s − 0.0361·69-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 0.54194133740.5419413374
L(12)L(\frac12) ≈\approx 0.54194133740.5419413374
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+0.486T+3T2 1 + 0.486T + 3T^{2}
7 1+3.63T+7T2 1 + 3.63T + 7T^{2}
11 1+2.79T+11T2 1 + 2.79T + 11T^{2}
13 1+2.86T+13T2 1 + 2.86T + 13T^{2}
17 1+1.17T+17T2 1 + 1.17T + 17T^{2}
23 1−0.617T+23T2 1 - 0.617T + 23T^{2}
29 1+4.96T+29T2 1 + 4.96T + 29T^{2}
31 1−0.745T+31T2 1 - 0.745T + 31T^{2}
37 1+8.23T+37T2 1 + 8.23T + 37T^{2}
41 1−9.98T+41T2 1 - 9.98T + 41T^{2}
43 1+10.4T+43T2 1 + 10.4T + 43T^{2}
47 1+5.07T+47T2 1 + 5.07T + 47T^{2}
53 1−7.45T+53T2 1 - 7.45T + 53T^{2}
59 1−3.83T+59T2 1 - 3.83T + 59T^{2}
61 1−11.2T+61T2 1 - 11.2T + 61T^{2}
67 1+6.10T+67T2 1 + 6.10T + 67T^{2}
71 1+9.40T+71T2 1 + 9.40T + 71T^{2}
73 1−9.52T+73T2 1 - 9.52T + 73T^{2}
79 1+3.70T+79T2 1 + 3.70T + 79T^{2}
83 1−4.66T+83T2 1 - 4.66T + 83T^{2}
89 1−10.6T+89T2 1 - 10.6T + 89T^{2}
97 1+0.629T+97T2 1 + 0.629T + 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

−8.593222436890973096530346120742, −7.67694376092315392620037092141, −6.97151306292891256872763844374, −6.26759576648153510219174166721, −5.51143912788659048075711805819, −4.93133487196376022304935597818, −3.69035037313820857998537092097, −2.98611152398659657223707538927, −2.20796277163223607968511295493, −0.39984940269767498431159631899, 0.39984940269767498431159631899, 2.20796277163223607968511295493, 2.98611152398659657223707538927, 3.69035037313820857998537092097, 4.93133487196376022304935597818, 5.51143912788659048075711805819, 6.26759576648153510219174166721, 6.97151306292891256872763844374, 7.67694376092315392620037092141, 8.593222436890973096530346120742

Graph of the ZZ-function along the critical line