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Properties

Label 2-2496-1.1-c3-0-46
Degree 22
Conductor 24962496
Sign 11
Analytic cond. 147.268147.268
Root an. cond. 12.135412.1354
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 1.46·5-s + 8.39·7-s + 9·9-s + 34.7·11-s + 13·13-s + 4.39·15-s − 108.·17-s + 143.·19-s − 25.1·21-s + 128.·23-s − 122.·25-s − 27·27-s + 18.8·29-s + 78.5·31-s − 104.·33-s − 12.2·35-s + 327.·37-s − 39·39-s + 327.·41-s − 336.·43-s − 13.1·45-s − 99.2·47-s − 272.·49-s + 324.·51-s + 686.·53-s − 50.9·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.130·5-s + 0.453·7-s + 0.333·9-s + 0.953·11-s + 0.277·13-s + 0.0756·15-s − 1.54·17-s + 1.72·19-s − 0.261·21-s + 1.16·23-s − 0.982·25-s − 0.192·27-s + 0.120·29-s + 0.455·31-s − 0.550·33-s − 0.0593·35-s + 1.45·37-s − 0.160·39-s + 1.24·41-s − 1.19·43-s − 0.0436·45-s − 0.308·47-s − 0.794·49-s + 0.890·51-s + 1.77·53-s − 0.124·55-s + ⋯

Functional equation

Λ(s)=(2496s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2496s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24962496    =    26⋅3⋅132^{6} \cdot 3 \cdot 13
Sign: 11
Analytic conductor: 147.268147.268
Root analytic conductor: 12.135412.1354
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2496, ( :3/2), 1)(2,\ 2496,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.1098112032.109811203
L(12)L(\frac12) ≈\approx 2.1098112032.109811203
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+3T 1 + 3T
13 1−13T 1 - 13T
good5 1+1.46T+125T2 1 + 1.46T + 125T^{2}
7 1−8.39T+343T2 1 - 8.39T + 343T^{2}
11 1−34.7T+1.33e3T2 1 - 34.7T + 1.33e3T^{2}
17 1+108.T+4.91e3T2 1 + 108.T + 4.91e3T^{2}
19 1−143.T+6.85e3T2 1 - 143.T + 6.85e3T^{2}
23 1−128.T+1.21e4T2 1 - 128.T + 1.21e4T^{2}
29 1−18.8T+2.43e4T2 1 - 18.8T + 2.43e4T^{2}
31 1−78.5T+2.97e4T2 1 - 78.5T + 2.97e4T^{2}
37 1−327.T+5.06e4T2 1 - 327.T + 5.06e4T^{2}
41 1−327.T+6.89e4T2 1 - 327.T + 6.89e4T^{2}
43 1+336.T+7.95e4T2 1 + 336.T + 7.95e4T^{2}
47 1+99.2T+1.03e5T2 1 + 99.2T + 1.03e5T^{2}
53 1−686.T+1.48e5T2 1 - 686.T + 1.48e5T^{2}
59 1+242.T+2.05e5T2 1 + 242.T + 2.05e5T^{2}
61 1−644.T+2.26e5T2 1 - 644.T + 2.26e5T^{2}
67 1+871.T+3.00e5T2 1 + 871.T + 3.00e5T^{2}
71 1+100.T+3.57e5T2 1 + 100.T + 3.57e5T^{2}
73 1−604.T+3.89e5T2 1 - 604.T + 3.89e5T^{2}
79 1+1.07e3T+4.93e5T2 1 + 1.07e3T + 4.93e5T^{2}
83 1−741.T+5.71e5T2 1 - 741.T + 5.71e5T^{2}
89 1+501.T+7.04e5T2 1 + 501.T + 7.04e5T^{2}
97 1+1.56e3T+9.12e5T2 1 + 1.56e3T + 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

−8.639134782559222820705833742138, −7.70914663507645320890912779024, −6.97813181518322587517449558663, −6.29379436037329359840412513949, −5.44435872815901686370455578666, −4.60547752360649103893513416751, −3.92425760569416123010242257170, −2.78466172469384241970553352331, −1.55136222901196661982519524271, −0.70686582964195287517465519461, 0.70686582964195287517465519461, 1.55136222901196661982519524271, 2.78466172469384241970553352331, 3.92425760569416123010242257170, 4.60547752360649103893513416751, 5.44435872815901686370455578666, 6.29379436037329359840412513949, 6.97813181518322587517449558663, 7.70914663507645320890912779024, 8.639134782559222820705833742138

Graph of the ZZ-function along the critical line