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Properties

Label 2-3744-1.1-c1-0-48
Degree 22
Conductor 37443744
Sign −1-1
Analytic cond. 29.895929.8959
Root an. cond. 5.467725.46772
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  − 0.622·5-s + 4.42·7-s − 5.80·11-s + 13-s − 2·17-s − 4.42·19-s + 8.85·23-s − 4.61·25-s − 2·29-s − 7.18·31-s − 2.75·35-s + 0.755·37-s − 3.37·41-s − 7.61·43-s − 1.80·47-s + 12.6·49-s − 4.75·53-s + 3.61·55-s − 11.0·59-s − 8.10·61-s − 0.622·65-s + 8.04·67-s + 7.05·71-s + 7.24·73-s − 25.7·77-s − 12·79-s + 3.05·83-s + ⋯
L(s)  = 1  − 0.278·5-s + 1.67·7-s − 1.75·11-s + 0.277·13-s − 0.485·17-s − 1.01·19-s + 1.84·23-s − 0.922·25-s − 0.371·29-s − 1.29·31-s − 0.465·35-s + 0.124·37-s − 0.527·41-s − 1.16·43-s − 0.263·47-s + 1.80·49-s − 0.653·53-s + 0.487·55-s − 1.43·59-s − 1.03·61-s − 0.0771·65-s + 0.982·67-s + 0.836·71-s + 0.847·73-s − 2.93·77-s − 1.35·79-s + 0.334·83-s + ⋯

Functional equation

Λ(s)=(3744s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3744s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37443744    =    25⋅32⋅132^{5} \cdot 3^{2} \cdot 13
Sign: −1-1
Analytic conductor: 29.895929.8959
Root analytic conductor: 5.467725.46772
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3744, ( :1/2), −1)(2,\ 3744,\ (\ :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
3 1 1
13 1−T 1 - T
good5 1+0.622T+5T2 1 + 0.622T + 5T^{2}
7 1−4.42T+7T2 1 - 4.42T + 7T^{2}
11 1+5.80T+11T2 1 + 5.80T + 11T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1+4.42T+19T2 1 + 4.42T + 19T^{2}
23 1−8.85T+23T2 1 - 8.85T + 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+7.18T+31T2 1 + 7.18T + 31T^{2}
37 1−0.755T+37T2 1 - 0.755T + 37T^{2}
41 1+3.37T+41T2 1 + 3.37T + 41T^{2}
43 1+7.61T+43T2 1 + 7.61T + 43T^{2}
47 1+1.80T+47T2 1 + 1.80T + 47T^{2}
53 1+4.75T+53T2 1 + 4.75T + 53T^{2}
59 1+11.0T+59T2 1 + 11.0T + 59T^{2}
61 1+8.10T+61T2 1 + 8.10T + 61T^{2}
67 1−8.04T+67T2 1 - 8.04T + 67T^{2}
71 1−7.05T+71T2 1 - 7.05T + 71T^{2}
73 1−7.24T+73T2 1 - 7.24T + 73T^{2}
79 1+12T+79T2 1 + 12T + 79T^{2}
83 1−3.05T+83T2 1 - 3.05T + 83T^{2}
89 1−1.86T+89T2 1 - 1.86T + 89T^{2}
97 1+0.755T+97T2 1 + 0.755T + 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

−8.101791113111504924053916294011, −7.61688404820186998899285648138, −6.82492143040815592550082672363, −5.68333500682587525358335729836, −5.01832010177739854468275385703, −4.56184701465775866478694369938, −3.44076801967697816800379304937, −2.36291370244336333494585776893, −1.57938427913110355421802480159, 0, 1.57938427913110355421802480159, 2.36291370244336333494585776893, 3.44076801967697816800379304937, 4.56184701465775866478694369938, 5.01832010177739854468275385703, 5.68333500682587525358335729836, 6.82492143040815592550082672363, 7.61688404820186998899285648138, 8.101791113111504924053916294011

Graph of the ZZ-function along the critical line