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Properties

Label 2-30400-1.1-c1-0-2
Degree 22
Conductor 3040030400
Sign 11
Analytic cond. 242.745242.745
Root an. cond. 15.580215.5802
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 3·9-s − 4·11-s − 2·13-s − 4·17-s + 19-s − 6·23-s + 6·29-s + 4·31-s − 10·37-s − 10·41-s − 2·43-s − 6·47-s − 3·49-s + 10·53-s − 2·61-s − 6·63-s − 8·67-s − 4·71-s − 4·73-s − 8·77-s − 4·79-s + 9·81-s + 18·83-s − 2·89-s − 4·91-s − 6·97-s + ⋯
L(s)  = 1  + 0.755·7-s − 9-s − 1.20·11-s − 0.554·13-s − 0.970·17-s + 0.229·19-s − 1.25·23-s + 1.11·29-s + 0.718·31-s − 1.64·37-s − 1.56·41-s − 0.304·43-s − 0.875·47-s − 3/7·49-s + 1.37·53-s − 0.256·61-s − 0.755·63-s − 0.977·67-s − 0.474·71-s − 0.468·73-s − 0.911·77-s − 0.450·79-s + 81-s + 1.97·83-s − 0.211·89-s − 0.419·91-s − 0.609·97-s + ⋯

Functional equation

Λ(s)=(30400s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(30400s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3040030400    =    26⋅52⋅192^{6} \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 242.745242.745
Root analytic conductor: 15.580215.5802
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 30400, ( :1/2), 1)(2,\ 30400,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.67842113420.6784211342
L(12)L(\frac12) ≈\approx 0.67842113420.6784211342
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+pT2 1 + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1+10T+pT2 1 + 10 T + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−18T+pT2 1 - 18 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1+6T+pT2 1 + 6 T + p T^{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

−15.10788360766588, −14.63591040162414, −13.88990011440446, −13.72705788477801, −13.13744925189553, −12.30442995308104, −11.84691704230534, −11.54303608190131, −10.79340721770672, −10.20063245167792, −10.04139016081147, −8.853056238810216, −8.658789712730202, −8.003228766947811, −7.641573015718220, −6.765778862114034, −6.276877088738288, −5.450457045467083, −5.007806169997759, −4.579193527580036, −3.607130834964076, −2.874743259551771, −2.295496907753385, −1.622291161487484, −0.2968376346866832, 0.2968376346866832, 1.622291161487484, 2.295496907753385, 2.874743259551771, 3.607130834964076, 4.579193527580036, 5.007806169997759, 5.450457045467083, 6.276877088738288, 6.765778862114034, 7.641573015718220, 8.003228766947811, 8.658789712730202, 8.853056238810216, 10.04139016081147, 10.20063245167792, 10.79340721770672, 11.54303608190131, 11.84691704230534, 12.30442995308104, 13.13744925189553, 13.72705788477801, 13.88990011440446, 14.63591040162414, 15.10788360766588

Graph of the ZZ-function along the critical line