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Properties

Label 2-78e2-1.1-c1-0-30
Degree 22
Conductor 60846084
Sign −1-1
Analytic cond. 48.580948.5809
Root an. cond. 6.970006.97000
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  − 3.04·5-s − 5.04·7-s + 3.80·11-s + 0.356·17-s − 6.13·19-s + 7.93·23-s + 4.29·25-s − 1.41·29-s + 5.78·31-s + 15.3·35-s + 0.0271·37-s + 9.18·41-s − 1.21·43-s + 9.56·47-s + 18.4·49-s + 4.73·53-s − 11.5·55-s − 13.5·59-s + 0.576·61-s − 7.95·67-s + 3.07·71-s − 14.0·73-s − 19.1·77-s − 0.841·79-s + 0.192·83-s − 1.08·85-s − 9.21·89-s + ⋯
L(s)  = 1  − 1.36·5-s − 1.90·7-s + 1.14·11-s + 0.0865·17-s − 1.40·19-s + 1.65·23-s + 0.859·25-s − 0.263·29-s + 1.03·31-s + 2.60·35-s + 0.00446·37-s + 1.43·41-s − 0.186·43-s + 1.39·47-s + 2.64·49-s + 0.650·53-s − 1.56·55-s − 1.76·59-s + 0.0738·61-s − 0.971·67-s + 0.365·71-s − 1.64·73-s − 2.18·77-s − 0.0946·79-s + 0.0211·83-s − 0.118·85-s − 0.976·89-s + ⋯

Functional equation

Λ(s)=(6084s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(6084s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 60846084    =    22⋅32⋅1322^{2} \cdot 3^{2} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 48.580948.5809
Root analytic conductor: 6.970006.97000
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 6084, ( :1/2), −1)(2,\ 6084,\ (\ :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 1
good5 1+3.04T+5T2 1 + 3.04T + 5T^{2}
7 1+5.04T+7T2 1 + 5.04T + 7T^{2}
11 1−3.80T+11T2 1 - 3.80T + 11T^{2}
17 1−0.356T+17T2 1 - 0.356T + 17T^{2}
19 1+6.13T+19T2 1 + 6.13T + 19T^{2}
23 1−7.93T+23T2 1 - 7.93T + 23T^{2}
29 1+1.41T+29T2 1 + 1.41T + 29T^{2}
31 1−5.78T+31T2 1 - 5.78T + 31T^{2}
37 1−0.0271T+37T2 1 - 0.0271T + 37T^{2}
41 1−9.18T+41T2 1 - 9.18T + 41T^{2}
43 1+1.21T+43T2 1 + 1.21T + 43T^{2}
47 1−9.56T+47T2 1 - 9.56T + 47T^{2}
53 1−4.73T+53T2 1 - 4.73T + 53T^{2}
59 1+13.5T+59T2 1 + 13.5T + 59T^{2}
61 1−0.576T+61T2 1 - 0.576T + 61T^{2}
67 1+7.95T+67T2 1 + 7.95T + 67T^{2}
71 1−3.07T+71T2 1 - 3.07T + 71T^{2}
73 1+14.0T+73T2 1 + 14.0T + 73T^{2}
79 1+0.841T+79T2 1 + 0.841T + 79T^{2}
83 1−0.192T+83T2 1 - 0.192T + 83T^{2}
89 1+9.21T+89T2 1 + 9.21T + 89T^{2}
97 1+0.682T+97T2 1 + 0.682T + 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

−7.53321824871438486671398148279, −6.99462821104612010901884328128, −6.42881477547968832529253141383, −5.81245891049363640066049011533, −4.43159022131840788035908704789, −4.06444510319973456994043443641, −3.27693864199093471942948575493, −2.65677668054898014742846885276, −0.997017769899252746805142115898, 0, 0.997017769899252746805142115898, 2.65677668054898014742846885276, 3.27693864199093471942948575493, 4.06444510319973456994043443641, 4.43159022131840788035908704789, 5.81245891049363640066049011533, 6.42881477547968832529253141383, 6.99462821104612010901884328128, 7.53321824871438486671398148279

Graph of the ZZ-function along the critical line