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Properties

Label 2-58800-1.1-c1-0-106
Degree 22
Conductor 5880058800
Sign −1-1
Analytic cond. 469.520469.520
Root an. cond. 21.668421.6684
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  − 3-s + 9-s − 6·11-s + 5·13-s − 6·17-s + 5·19-s − 6·23-s − 27-s − 6·29-s − 31-s + 6·33-s − 2·37-s − 5·39-s − 43-s − 6·47-s + 6·51-s + 12·53-s − 5·57-s − 6·59-s + 13·61-s + 11·67-s + 6·69-s + 2·73-s − 8·79-s + 81-s + 6·83-s + 6·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s + 1.38·13-s − 1.45·17-s + 1.14·19-s − 1.25·23-s − 0.192·27-s − 1.11·29-s − 0.179·31-s + 1.04·33-s − 0.328·37-s − 0.800·39-s − 0.152·43-s − 0.875·47-s + 0.840·51-s + 1.64·53-s − 0.662·57-s − 0.781·59-s + 1.66·61-s + 1.34·67-s + 0.722·69-s + 0.234·73-s − 0.900·79-s + 1/9·81-s + 0.658·83-s + 0.643·87-s + ⋯

Functional equation

Λ(s)=(58800s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(58800s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5880058800    =    24⋅3⋅52⋅722^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 469.520469.520
Root analytic conductor: 21.668421.6684
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 58800, ( :1/2), −1)(2,\ 58800,\ (\ :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+T 1 + T
5 1 1
7 1 1
good11 1+6T+pT2 1 + 6 T + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−5T+pT2 1 - 5 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+T+pT2 1 + T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−13T+pT2 1 - 13 T + p T^{2}
67 1−11T+pT2 1 - 11 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+7T+pT2 1 + 7 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

−14.56408312303212, −13.94213956873909, −13.28472936323399, −13.25937022594090, −12.72837353274949, −11.89063581939062, −11.53490097197637, −10.91516978279511, −10.72000961802173, −10.01415924908824, −9.613227637066306, −8.820025413364233, −8.344538929216669, −7.851416658571146, −7.256593655407632, −6.691434088566023, −6.053573462854549, −5.479320955394333, −5.214549328857454, −4.379849736484535, −3.801510771022759, −3.178753328126813, −2.291042289879081, −1.820468025319226, −0.7612954584705766, 0, 0.7612954584705766, 1.820468025319226, 2.291042289879081, 3.178753328126813, 3.801510771022759, 4.379849736484535, 5.214549328857454, 5.479320955394333, 6.053573462854549, 6.691434088566023, 7.256593655407632, 7.851416658571146, 8.344538929216669, 8.820025413364233, 9.613227637066306, 10.01415924908824, 10.72000961802173, 10.91516978279511, 11.53490097197637, 11.89063581939062, 12.72837353274949, 13.25937022594090, 13.28472936323399, 13.94213956873909, 14.56408312303212

Graph of the ZZ-function along the critical line