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Properties

Label 2-3960-1.1-c1-0-2
Degree 22
Conductor 39603960
Sign 11
Analytic cond. 31.620731.6207
Root an. cond. 5.623235.62323
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 − 5.12·7-s − 11-s − 3.12·13-s − 3.12·17-s + 4·23-s + 25-s − 2·29-s − 5.12·35-s + 6·37-s − 6·41-s + 5.12·43-s − 4·47-s + 19.2·49-s + 8.24·53-s − 55-s − 4·59-s + 10·61-s − 3.12·65-s − 6.24·67-s + 6.24·71-s + 4.87·73-s + 5.12·77-s + 2.24·79-s − 11.3·83-s − 3.12·85-s + 16.2·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.93·7-s − 0.301·11-s − 0.866·13-s − 0.757·17-s + 0.834·23-s + 0.200·25-s − 0.371·29-s − 0.865·35-s + 0.986·37-s − 0.937·41-s + 0.781·43-s − 0.583·47-s + 2.74·49-s + 1.13·53-s − 0.134·55-s − 0.520·59-s + 1.28·61-s − 0.387·65-s − 0.763·67-s + 0.741·71-s + 0.570·73-s + 0.583·77-s + 0.252·79-s − 1.24·83-s − 0.338·85-s + 1.72·89-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 11
Analytic conductor: 31.620731.6207
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3960, ( :1/2), 1)(2,\ 3960,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1035663161.103566316
L(12)L(\frac12) ≈\approx 1.1035663161.103566316
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
11 1+T 1 + T
good7 1+5.12T+7T2 1 + 5.12T + 7T^{2}
13 1+3.12T+13T2 1 + 3.12T + 13T^{2}
17 1+3.12T+17T2 1 + 3.12T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−6T+37T2 1 - 6T + 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−5.12T+43T2 1 - 5.12T + 43T^{2}
47 1+4T+47T2 1 + 4T + 47T^{2}
53 1−8.24T+53T2 1 - 8.24T + 53T^{2}
59 1+4T+59T2 1 + 4T + 59T^{2}
61 1−10T+61T2 1 - 10T + 61T^{2}
67 1+6.24T+67T2 1 + 6.24T + 67T^{2}
71 1−6.24T+71T2 1 - 6.24T + 71T^{2}
73 1−4.87T+73T2 1 - 4.87T + 73T^{2}
79 1−2.24T+79T2 1 - 2.24T + 79T^{2}
83 1+11.3T+83T2 1 + 11.3T + 83T^{2}
89 1−16.2T+89T2 1 - 16.2T + 89T^{2}
97 1−12.2T+97T2 1 - 12.2T + 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

−8.681927373356142504717409803471, −7.52444327848299020739194379237, −6.90183151129377935470332399835, −6.33194746375446376072284713970, −5.59790009145838515672466575544, −4.73108754265234899709649356983, −3.71360024776516049312484128673, −2.89622172001196878056084332535, −2.22995185898318267104759345975, −0.56887589256857787452940651825, 0.56887589256857787452940651825, 2.22995185898318267104759345975, 2.89622172001196878056084332535, 3.71360024776516049312484128673, 4.73108754265234899709649356983, 5.59790009145838515672466575544, 6.33194746375446376072284713970, 6.90183151129377935470332399835, 7.52444327848299020739194379237, 8.681927373356142504717409803471

Graph of the ZZ-function along the critical line