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Properties

Label 2-968-1.1-c1-0-17
Degree 22
Conductor 968968
Sign −1-1
Analytic cond. 7.729517.72951
Root an. cond. 2.780202.78020
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  − 1.61·3-s + 0.618·5-s − 1.61·7-s − 0.381·9-s + 5.85·13-s − 1.00·15-s + 1.85·17-s − 4.85·19-s + 2.61·21-s − 4·23-s − 4.61·25-s + 5.47·27-s − 7.32·29-s + 1.09·31-s − 1.00·35-s − 9.61·37-s − 9.47·39-s + 9.61·41-s − 1.52·43-s − 0.236·45-s − 10.5·47-s − 4.38·49-s − 3·51-s + 0.618·53-s + 7.85·57-s − 2.38·59-s + 1.85·61-s + ⋯
L(s)  = 1  − 0.934·3-s + 0.276·5-s − 0.611·7-s − 0.127·9-s + 1.62·13-s − 0.258·15-s + 0.449·17-s − 1.11·19-s + 0.571·21-s − 0.834·23-s − 0.923·25-s + 1.05·27-s − 1.36·29-s + 0.195·31-s − 0.169·35-s − 1.58·37-s − 1.51·39-s + 1.50·41-s − 0.232·43-s − 0.0351·45-s − 1.54·47-s − 0.625·49-s − 0.420·51-s + 0.0848·53-s + 1.04·57-s − 0.310·59-s + 0.237·61-s + ⋯

Functional equation

Λ(s)=(968s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(968s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    23⋅1122^{3} \cdot 11^{2}
Sign: −1-1
Analytic conductor: 7.729517.72951
Root analytic conductor: 2.780202.78020
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 968, ( :1/2), −1)(2,\ 968,\ (\ :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
11 1 1
good3 1+1.61T+3T2 1 + 1.61T + 3T^{2}
5 1−0.618T+5T2 1 - 0.618T + 5T^{2}
7 1+1.61T+7T2 1 + 1.61T + 7T^{2}
13 1−5.85T+13T2 1 - 5.85T + 13T^{2}
17 1−1.85T+17T2 1 - 1.85T + 17T^{2}
19 1+4.85T+19T2 1 + 4.85T + 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+7.32T+29T2 1 + 7.32T + 29T^{2}
31 1−1.09T+31T2 1 - 1.09T + 31T^{2}
37 1+9.61T+37T2 1 + 9.61T + 37T^{2}
41 1−9.61T+41T2 1 - 9.61T + 41T^{2}
43 1+1.52T+43T2 1 + 1.52T + 43T^{2}
47 1+10.5T+47T2 1 + 10.5T + 47T^{2}
53 1−0.618T+53T2 1 - 0.618T + 53T^{2}
59 1+2.38T+59T2 1 + 2.38T + 59T^{2}
61 1−1.85T+61T2 1 - 1.85T + 61T^{2}
67 1+14.4T+67T2 1 + 14.4T + 67T^{2}
71 1+5.09T+71T2 1 + 5.09T + 71T^{2}
73 1−3.14T+73T2 1 - 3.14T + 73T^{2}
79 1−3.85T+79T2 1 - 3.85T + 79T^{2}
83 1+9.38T+83T2 1 + 9.38T + 83T^{2}
89 1+4.47T+89T2 1 + 4.47T + 89T^{2}
97 1+12.3T+97T2 1 + 12.3T + 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

−9.718155373631337909562928826012, −8.790340375267244736428367243560, −7.993669350098141213010103018783, −6.72500884991025223316708225768, −6.02617977199967583271809473827, −5.59912022531615973356578863948, −4.23320561779896701245064714031, −3.28203777656175467468818893269, −1.70035795728857034723097312919, 0, 1.70035795728857034723097312919, 3.28203777656175467468818893269, 4.23320561779896701245064714031, 5.59912022531615973356578863948, 6.02617977199967583271809473827, 6.72500884991025223316708225768, 7.993669350098141213010103018783, 8.790340375267244736428367243560, 9.718155373631337909562928826012

Graph of the ZZ-function along the critical line