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Properties

Label 2-6080-1.1-c1-0-107
Degree 22
Conductor 60806080
Sign −1-1
Analytic cond. 48.549048.5490
Root an. cond. 6.967716.96771
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  − 1.41·3-s + 5-s + 2.82·7-s − 0.999·9-s − 0.828·11-s − 3.41·13-s − 1.41·15-s + 4.82·17-s − 19-s − 4.00·21-s − 4·23-s + 25-s + 5.65·27-s + 0.828·29-s + 1.17·33-s + 2.82·35-s − 10.2·37-s + 4.82·39-s − 0.828·41-s + 2.82·43-s − 0.999·45-s + 8.48·47-s + 1.00·49-s − 6.82·51-s − 13.0·53-s − 0.828·55-s + 1.41·57-s + ⋯
L(s)  = 1  − 0.816·3-s + 0.447·5-s + 1.06·7-s − 0.333·9-s − 0.249·11-s − 0.946·13-s − 0.365·15-s + 1.17·17-s − 0.229·19-s − 0.872·21-s − 0.834·23-s + 0.200·25-s + 1.08·27-s + 0.153·29-s + 0.203·33-s + 0.478·35-s − 1.68·37-s + 0.773·39-s − 0.129·41-s + 0.431·43-s − 0.149·45-s + 1.23·47-s + 0.142·49-s − 0.956·51-s − 1.79·53-s − 0.111·55-s + 0.187·57-s + ⋯

Functional equation

Λ(s)=(6080s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(6080s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 60806080    =    26⋅5⋅192^{6} \cdot 5 \cdot 19
Sign: −1-1
Analytic conductor: 48.549048.5490
Root analytic conductor: 6.967716.96771
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 6080, ( :1/2), −1)(2,\ 6080,\ (\ :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
5 1−T 1 - T
19 1+T 1 + T
good3 1+1.41T+3T2 1 + 1.41T + 3T^{2}
7 1−2.82T+7T2 1 - 2.82T + 7T^{2}
11 1+0.828T+11T2 1 + 0.828T + 11T^{2}
13 1+3.41T+13T2 1 + 3.41T + 13T^{2}
17 1−4.82T+17T2 1 - 4.82T + 17T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−0.828T+29T2 1 - 0.828T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+10.2T+37T2 1 + 10.2T + 37T^{2}
41 1+0.828T+41T2 1 + 0.828T + 41T^{2}
43 1−2.82T+43T2 1 - 2.82T + 43T^{2}
47 1−8.48T+47T2 1 - 8.48T + 47T^{2}
53 1+13.0T+53T2 1 + 13.0T + 53T^{2}
59 1−2.82T+59T2 1 - 2.82T + 59T^{2}
61 1−1.65T+61T2 1 - 1.65T + 61T^{2}
67 1+9.41T+67T2 1 + 9.41T + 67T^{2}
71 1+15.3T+71T2 1 + 15.3T + 71T^{2}
73 1−12.1T+73T2 1 - 12.1T + 73T^{2}
79 1−9.17T+79T2 1 - 9.17T + 79T^{2}
83 1−8T+83T2 1 - 8T + 83T^{2}
89 1+3.17T+89T2 1 + 3.17T + 89T^{2}
97 1+2.24T+97T2 1 + 2.24T + 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

−7.78513685174352828177882674767, −6.97069244630425161000059216205, −6.15017307823796868359685993137, −5.41413550682614316386569133086, −5.11120029313900633291150555543, −4.28059413026400237982553779563, −3.13566745663642548051411453896, −2.20660523819540497949679362949, −1.29501968330739158341611946759, 0, 1.29501968330739158341611946759, 2.20660523819540497949679362949, 3.13566745663642548051411453896, 4.28059413026400237982553779563, 5.11120029313900633291150555543, 5.41413550682614316386569133086, 6.15017307823796868359685993137, 6.97069244630425161000059216205, 7.78513685174352828177882674767

Graph of the ZZ-function along the critical line