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Properties

Label 2-9610-1.1-c1-0-35
Degree 22
Conductor 96109610
Sign 11
Analytic cond. 76.736276.7362
Root an. cond. 8.759928.75992
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 4.23·7-s − 8-s − 2·9-s − 10-s + 3·11-s + 12-s + 0.236·13-s + 4.23·14-s + 15-s + 16-s − 1.14·17-s + 2·18-s − 6.47·19-s + 20-s − 4.23·21-s − 3·22-s + 1.14·23-s − 24-s + 25-s − 0.236·26-s − 5·27-s − 4.23·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.60·7-s − 0.353·8-s − 0.666·9-s − 0.316·10-s + 0.904·11-s + 0.288·12-s + 0.0654·13-s + 1.13·14-s + 0.258·15-s + 0.250·16-s − 0.277·17-s + 0.471·18-s − 1.48·19-s + 0.223·20-s − 0.924·21-s − 0.639·22-s + 0.238·23-s − 0.204·24-s + 0.200·25-s − 0.0462·26-s − 0.962·27-s − 0.800·28-s + ⋯

Functional equation

Λ(s)=(9610s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9610s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 96109610    =    2⋅5⋅3122 \cdot 5 \cdot 31^{2}
Sign: 11
Analytic conductor: 76.736276.7362
Root analytic conductor: 8.759928.75992
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9610, ( :1/2), 1)(2,\ 9610,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.99312257300.9931225730
L(12)L(\frac12) ≈\approx 0.99312257300.9931225730
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+T 1 + T
5 1−T 1 - T
31 1 1
good3 1−T+3T2 1 - T + 3T^{2}
7 1+4.23T+7T2 1 + 4.23T + 7T^{2}
11 1−3T+11T2 1 - 3T + 11T^{2}
13 1−0.236T+13T2 1 - 0.236T + 13T^{2}
17 1+1.14T+17T2 1 + 1.14T + 17T^{2}
19 1+6.47T+19T2 1 + 6.47T + 19T^{2}
23 1−1.14T+23T2 1 - 1.14T + 23T^{2}
29 1+3T+29T2 1 + 3T + 29T^{2}
37 1+7.94T+37T2 1 + 7.94T + 37T^{2}
41 1−10.8T+41T2 1 - 10.8T + 41T^{2}
43 1+1.23T+43T2 1 + 1.23T + 43T^{2}
47 1+9T+47T2 1 + 9T + 47T^{2}
53 1+13.4T+53T2 1 + 13.4T + 53T^{2}
59 1−6T+59T2 1 - 6T + 59T^{2}
61 1+1.61T+61T2 1 + 1.61T + 61T^{2}
67 1−14.7T+67T2 1 - 14.7T + 67T^{2}
71 1+1.85T+71T2 1 + 1.85T + 71T^{2}
73 1−12.1T+73T2 1 - 12.1T + 73T^{2}
79 1−4.38T+79T2 1 - 4.38T + 79T^{2}
83 1−0.708T+83T2 1 - 0.708T + 83T^{2}
89 1−12T+89T2 1 - 12T + 89T^{2}
97 1−3.23T+97T2 1 - 3.23T + 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

−7.78691393395075565987888678837, −6.92160161182102774993918191113, −6.30642692930879235590851674865, −6.12498319989366140345176256922, −5.00234957259385455347198496386, −3.80177707837177269023637324294, −3.37655978446979920846907163676, −2.49617496702618766571630428197, −1.85651769414923188852093045253, −0.48555217773495951058936874804, 0.48555217773495951058936874804, 1.85651769414923188852093045253, 2.49617496702618766571630428197, 3.37655978446979920846907163676, 3.80177707837177269023637324294, 5.00234957259385455347198496386, 6.12498319989366140345176256922, 6.30642692930879235590851674865, 6.92160161182102774993918191113, 7.78691393395075565987888678837

Graph of the ZZ-function along the critical line