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Properties

Label 2-608-1.1-c1-0-14
Degree 22
Conductor 608608
Sign −1-1
Analytic cond. 4.854904.85490
Root an. cond. 2.203382.20338
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 3·9-s + 3·11-s − 4·13-s − 3·17-s + 19-s − 8·23-s − 4·25-s + 2·31-s + 35-s − 8·37-s + 11·43-s + 3·45-s − 7·47-s − 6·49-s + 2·53-s − 3·55-s + 6·59-s − 61-s + 3·63-s + 4·65-s − 10·67-s + 2·71-s + 5·73-s − 3·77-s − 2·79-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 9-s + 0.904·11-s − 1.10·13-s − 0.727·17-s + 0.229·19-s − 1.66·23-s − 4/5·25-s + 0.359·31-s + 0.169·35-s − 1.31·37-s + 1.67·43-s + 0.447·45-s − 1.02·47-s − 6/7·49-s + 0.274·53-s − 0.404·55-s + 0.781·59-s − 0.128·61-s + 0.377·63-s + 0.496·65-s − 1.22·67-s + 0.237·71-s + 0.585·73-s − 0.341·77-s − 0.225·79-s + ⋯

Functional equation

Λ(s)=(608s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(608s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 608608    =    25⋅192^{5} \cdot 19
Sign: −1-1
Analytic conductor: 4.854904.85490
Root analytic conductor: 2.203382.20338
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 608, ( :1/2), −1)(2,\ 608,\ (\ :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
19 1−T 1 - T
good3 1+pT2 1 + p T^{2}
5 1+T+pT2 1 + T + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−11T+pT2 1 - 11 T + p T^{2}
47 1+7T+pT2 1 + 7 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+T+pT2 1 + T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1−2T+pT2 1 - 2 T + p T^{2}
73 1−5T+pT2 1 - 5 T + p T^{2}
79 1+2T+pT2 1 + 2 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+12T+pT2 1 + 12 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

−10.11217440262524272793966315283, −9.356073062840077109182609673978, −8.467317306890464867116303654123, −7.59019329531238695962577712799, −6.57705024982005416568601729042, −5.71435981136868716408885186688, −4.47863079866238157165170735095, −3.47456597646201355004490006149, −2.19568966813265176024980758112, 0, 2.19568966813265176024980758112, 3.47456597646201355004490006149, 4.47863079866238157165170735095, 5.71435981136868716408885186688, 6.57705024982005416568601729042, 7.59019329531238695962577712799, 8.467317306890464867116303654123, 9.356073062840077109182609673978, 10.11217440262524272793966315283

Graph of the ZZ-function along the critical line