Properties

Label 2-6660-1.1-c1-0-32
Degree 22
Conductor 66606660
Sign 11
Analytic cond. 53.180353.1803
Root an. cond. 7.292487.29248
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 + 1.26·7-s + 5.46·11-s + 4·13-s + 4·17-s + 6.19·19-s + 8.92·23-s + 25-s − 7.46·29-s + 10.1·31-s − 1.26·35-s + 37-s − 4.92·41-s + 6·43-s − 1.26·47-s − 5.39·49-s − 12.9·53-s − 5.46·55-s − 3.66·59-s + 14·61-s − 4·65-s − 5.26·67-s − 8·71-s − 10.3·73-s + 6.92·77-s + 6.19·79-s − 1.66·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.479·7-s + 1.64·11-s + 1.10·13-s + 0.970·17-s + 1.42·19-s + 1.86·23-s + 0.200·25-s − 1.38·29-s + 1.83·31-s − 0.214·35-s + 0.164·37-s − 0.769·41-s + 0.914·43-s − 0.184·47-s − 0.770·49-s − 1.77·53-s − 0.736·55-s − 0.476·59-s + 1.79·61-s − 0.496·65-s − 0.643·67-s − 0.949·71-s − 1.21·73-s + 0.789·77-s + 0.697·79-s − 0.182·83-s + ⋯

Functional equation

Λ(s)=(6660s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6660s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 66606660    =    22325372^{2} \cdot 3^{2} \cdot 5 \cdot 37
Sign: 11
Analytic conductor: 53.180353.1803
Root analytic conductor: 7.292487.29248
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6660, ( :1/2), 1)(2,\ 6660,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) \approx 2.8505638652.850563865
L(12)L(\frac12) \approx 2.8505638652.850563865
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)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
37 1T 1 - T
good7 11.26T+7T2 1 - 1.26T + 7T^{2}
11 15.46T+11T2 1 - 5.46T + 11T^{2}
13 14T+13T2 1 - 4T + 13T^{2}
17 14T+17T2 1 - 4T + 17T^{2}
19 16.19T+19T2 1 - 6.19T + 19T^{2}
23 18.92T+23T2 1 - 8.92T + 23T^{2}
29 1+7.46T+29T2 1 + 7.46T + 29T^{2}
31 110.1T+31T2 1 - 10.1T + 31T^{2}
41 1+4.92T+41T2 1 + 4.92T + 41T^{2}
43 16T+43T2 1 - 6T + 43T^{2}
47 1+1.26T+47T2 1 + 1.26T + 47T^{2}
53 1+12.9T+53T2 1 + 12.9T + 53T^{2}
59 1+3.66T+59T2 1 + 3.66T + 59T^{2}
61 114T+61T2 1 - 14T + 61T^{2}
67 1+5.26T+67T2 1 + 5.26T + 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+10.3T+73T2 1 + 10.3T + 73T^{2}
79 16.19T+79T2 1 - 6.19T + 79T^{2}
83 1+1.66T+83T2 1 + 1.66T + 83T^{2}
89 12T+89T2 1 - 2T + 89T^{2}
97 1+14T+97T2 1 + 14T + 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−7.985796035268412574264512571425, −7.30003461503997691114460293543, −6.64194599837109156851624099607, −5.90989875910007022026534618012, −5.11063074685569888081893924172, −4.34665405293149061196932876670, −3.51683620110618350610943882838, −3.04029320729344446829243626781, −1.40885739769473434892341782349, −1.06571257106687043127735882754, 1.06571257106687043127735882754, 1.40885739769473434892341782349, 3.04029320729344446829243626781, 3.51683620110618350610943882838, 4.34665405293149061196932876670, 5.11063074685569888081893924172, 5.90989875910007022026534618012, 6.64194599837109156851624099607, 7.30003461503997691114460293543, 7.985796035268412574264512571425

Graph of the ZZ-function along the critical line