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Properties

Label 2-4140-1.1-c1-0-1
Degree 22
Conductor 41404140
Sign 11
Analytic cond. 33.058033.0580
Root an. cond. 5.749615.74961
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  − 5-s − 2.46·7-s − 4.19·11-s − 3.26·13-s + 7.73·17-s + 0.732·19-s + 23-s + 25-s − 7.19·29-s − 31-s + 2.46·35-s − 11.3·37-s + 7.73·41-s + 3.46·43-s − 0.732·47-s − 0.928·49-s − 6.66·53-s + 4.19·55-s + 7.19·59-s − 10.7·61-s + 3.26·65-s + 5·67-s + 14.1·71-s − 11.2·73-s + 10.3·77-s + 4·79-s + 6.66·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.931·7-s − 1.26·11-s − 0.906·13-s + 1.87·17-s + 0.167·19-s + 0.208·23-s + 0.200·25-s − 1.33·29-s − 0.179·31-s + 0.416·35-s − 1.87·37-s + 1.20·41-s + 0.528·43-s − 0.106·47-s − 0.132·49-s − 0.914·53-s + 0.565·55-s + 0.936·59-s − 1.37·61-s + 0.405·65-s + 0.610·67-s + 1.67·71-s − 1.31·73-s + 1.17·77-s + 0.450·79-s + 0.731·83-s + ⋯

Functional equation

Λ(s)=(4140s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4140s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 41404140    =    22⋅32⋅5⋅232^{2} \cdot 3^{2} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 33.058033.0580
Root analytic conductor: 5.749615.74961
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4140, ( :1/2), 1)(2,\ 4140,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.97664430160.9766443016
L(12)L(\frac12) ≈\approx 0.97664430160.9766443016
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
5 1+T 1 + T
23 1−T 1 - T
good7 1+2.46T+7T2 1 + 2.46T + 7T^{2}
11 1+4.19T+11T2 1 + 4.19T + 11T^{2}
13 1+3.26T+13T2 1 + 3.26T + 13T^{2}
17 1−7.73T+17T2 1 - 7.73T + 17T^{2}
19 1−0.732T+19T2 1 - 0.732T + 19T^{2}
29 1+7.19T+29T2 1 + 7.19T + 29T^{2}
31 1+T+31T2 1 + T + 31T^{2}
37 1+11.3T+37T2 1 + 11.3T + 37T^{2}
41 1−7.73T+41T2 1 - 7.73T + 41T^{2}
43 1−3.46T+43T2 1 - 3.46T + 43T^{2}
47 1+0.732T+47T2 1 + 0.732T + 47T^{2}
53 1+6.66T+53T2 1 + 6.66T + 53T^{2}
59 1−7.19T+59T2 1 - 7.19T + 59T^{2}
61 1+10.7T+61T2 1 + 10.7T + 61T^{2}
67 1−5T+67T2 1 - 5T + 67T^{2}
71 1−14.1T+71T2 1 - 14.1T + 71T^{2}
73 1+11.2T+73T2 1 + 11.2T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−6.66T+83T2 1 - 6.66T + 83T^{2}
89 1−16.3T+89T2 1 - 16.3T + 89T^{2}
97 1−4T+97T2 1 - 4T + 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

−8.271666437361429482486957972676, −7.48471910346565190155131549742, −7.31013820306469509939203608450, −6.11824043870575387337461853206, −5.42390838381168569674357040944, −4.80811105984273144911628066603, −3.53442423262008260350546858260, −3.15549718158090759245168180559, −2.07921534633835402326857337660, −0.53604838269124823699051315911, 0.53604838269124823699051315911, 2.07921534633835402326857337660, 3.15549718158090759245168180559, 3.53442423262008260350546858260, 4.80811105984273144911628066603, 5.42390838381168569674357040944, 6.11824043870575387337461853206, 7.31013820306469509939203608450, 7.48471910346565190155131549742, 8.271666437361429482486957972676

Graph of the ZZ-function along the critical line