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Properties

Label 2-48e2-1.1-c3-0-100
Degree 22
Conductor 23042304
Sign −1-1
Analytic cond. 135.940135.940
Root an. cond. 11.659311.6593
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s + 21.1·7-s + 42.3·11-s + 20·13-s + 8·17-s − 84.6·19-s − 169.·23-s − 89·25-s + 46·29-s + 21.1·31-s − 126.·35-s − 164·37-s + 312·41-s − 423.·43-s − 169.·47-s + 105.·49-s − 266·53-s − 253.·55-s − 253.·59-s − 132·61-s − 120·65-s − 507.·67-s + 677.·71-s + 246·73-s + 896.·77-s − 232.·79-s + 973.·83-s + ⋯
L(s)  = 1  − 0.536·5-s + 1.14·7-s + 1.16·11-s + 0.426·13-s + 0.114·17-s − 1.02·19-s − 1.53·23-s − 0.711·25-s + 0.294·29-s + 0.122·31-s − 0.613·35-s − 0.728·37-s + 1.18·41-s − 1.50·43-s − 0.525·47-s + 0.306·49-s − 0.689·53-s − 0.622·55-s − 0.560·59-s − 0.277·61-s − 0.228·65-s − 0.926·67-s + 1.13·71-s + 0.394·73-s + 1.32·77-s − 0.331·79-s + 1.28·83-s + ⋯

Functional equation

Λ(s)=(2304s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(2304s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23042304    =    28⋅322^{8} \cdot 3^{2}
Sign: −1-1
Analytic conductor: 135.940135.940
Root analytic conductor: 11.659311.6593
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2304, ( :3/2), −1)(2,\ 2304,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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 1
good5 1+6T+125T2 1 + 6T + 125T^{2}
7 1−21.1T+343T2 1 - 21.1T + 343T^{2}
11 1−42.3T+1.33e3T2 1 - 42.3T + 1.33e3T^{2}
13 1−20T+2.19e3T2 1 - 20T + 2.19e3T^{2}
17 1−8T+4.91e3T2 1 - 8T + 4.91e3T^{2}
19 1+84.6T+6.85e3T2 1 + 84.6T + 6.85e3T^{2}
23 1+169.T+1.21e4T2 1 + 169.T + 1.21e4T^{2}
29 1−46T+2.43e4T2 1 - 46T + 2.43e4T^{2}
31 1−21.1T+2.97e4T2 1 - 21.1T + 2.97e4T^{2}
37 1+164T+5.06e4T2 1 + 164T + 5.06e4T^{2}
41 1−312T+6.89e4T2 1 - 312T + 6.89e4T^{2}
43 1+423.T+7.95e4T2 1 + 423.T + 7.95e4T^{2}
47 1+169.T+1.03e5T2 1 + 169.T + 1.03e5T^{2}
53 1+266T+1.48e5T2 1 + 266T + 1.48e5T^{2}
59 1+253.T+2.05e5T2 1 + 253.T + 2.05e5T^{2}
61 1+132T+2.26e5T2 1 + 132T + 2.26e5T^{2}
67 1+507.T+3.00e5T2 1 + 507.T + 3.00e5T^{2}
71 1−677.T+3.57e5T2 1 - 677.T + 3.57e5T^{2}
73 1−246T+3.89e5T2 1 - 246T + 3.89e5T^{2}
79 1+232.T+4.93e5T2 1 + 232.T + 4.93e5T^{2}
83 1−973.T+5.71e5T2 1 - 973.T + 5.71e5T^{2}
89 1−1.39e3T+7.04e5T2 1 - 1.39e3T + 7.04e5T^{2}
97 1+302T+9.12e5T2 1 + 302T + 9.12e5T^{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

−8.151983544079724472110008609149, −7.77092133738538295394211340733, −6.62444761569389701250182031264, −6.05368044855299303696892849446, −4.92522210563609866280458026059, −4.17708255379714852694468200092, −3.57012330100525943800995470349, −2.09107140383675182861094831733, −1.35637473763195318488072520121, 0, 1.35637473763195318488072520121, 2.09107140383675182861094831733, 3.57012330100525943800995470349, 4.17708255379714852694468200092, 4.92522210563609866280458026059, 6.05368044855299303696892849446, 6.62444761569389701250182031264, 7.77092133738538295394211340733, 8.151983544079724472110008609149

Graph of the ZZ-function along the critical line