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Properties

Label 2-9610-1.1-c1-0-132
Degree 22
Conductor 96109610
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.84·3-s + 4-s − 5-s − 1.84·6-s − 5.01·7-s + 8-s + 0.414·9-s − 10-s − 4.69·11-s − 1.84·12-s + 3.83·13-s − 5.01·14-s + 1.84·15-s + 16-s − 5.78·17-s + 0.414·18-s + 5.19·19-s − 20-s + 9.25·21-s − 4.69·22-s + 6.40·23-s − 1.84·24-s + 25-s + 3.83·26-s + 4.77·27-s − 5.01·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.06·3-s + 0.5·4-s − 0.447·5-s − 0.754·6-s − 1.89·7-s + 0.353·8-s + 0.138·9-s − 0.316·10-s − 1.41·11-s − 0.533·12-s + 1.06·13-s − 1.33·14-s + 0.477·15-s + 0.250·16-s − 1.40·17-s + 0.0976·18-s + 1.19·19-s − 0.223·20-s + 2.02·21-s − 1.00·22-s + 1.33·23-s − 0.377·24-s + 0.200·25-s + 0.752·26-s + 0.919·27-s − 0.946·28-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: −1-1
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: 11
Selberg data: (2, 9610, ( :1/2), −1)(2,\ 9610,\ (\ :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−T 1 - T
5 1+T 1 + T
31 1 1
good3 1+1.84T+3T2 1 + 1.84T + 3T^{2}
7 1+5.01T+7T2 1 + 5.01T + 7T^{2}
11 1+4.69T+11T2 1 + 4.69T + 11T^{2}
13 1−3.83T+13T2 1 - 3.83T + 13T^{2}
17 1+5.78T+17T2 1 + 5.78T + 17T^{2}
19 1−5.19T+19T2 1 - 5.19T + 19T^{2}
23 1−6.40T+23T2 1 - 6.40T + 23T^{2}
29 1−5.01T+29T2 1 - 5.01T + 29T^{2}
37 1−1.86T+37T2 1 - 1.86T + 37T^{2}
41 1+7.74T+41T2 1 + 7.74T + 41T^{2}
43 1−0.904T+43T2 1 - 0.904T + 43T^{2}
47 1−0.257T+47T2 1 - 0.257T + 47T^{2}
53 1+3.83T+53T2 1 + 3.83T + 53T^{2}
59 1−4.16T+59T2 1 - 4.16T + 59T^{2}
61 1+2.99T+61T2 1 + 2.99T + 61T^{2}
67 1+1.92T+67T2 1 + 1.92T + 67T^{2}
71 1+5.17T+71T2 1 + 5.17T + 71T^{2}
73 1+1.17T+73T2 1 + 1.17T + 73T^{2}
79 1−14.8T+79T2 1 - 14.8T + 79T^{2}
83 1−14.9T+83T2 1 - 14.9T + 83T^{2}
89 1−13.5T+89T2 1 - 13.5T + 89T^{2}
97 1+8.36T+97T2 1 + 8.36T + 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

−6.94537774491825474321855022161, −6.53398540560926640410736908268, −6.04360775327742847174187576290, −5.24068476672548678858500091700, −4.81688824701845601627966015169, −3.75064038521084321737536799683, −3.10447262354289391171114300363, −2.58824464373040444120139441430, −0.903409693238353169384284592919, 0, 0.903409693238353169384284592919, 2.58824464373040444120139441430, 3.10447262354289391171114300363, 3.75064038521084321737536799683, 4.81688824701845601627966015169, 5.24068476672548678858500091700, 6.04360775327742847174187576290, 6.53398540560926640410736908268, 6.94537774491825474321855022161

Graph of the ZZ-function along the critical line