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Properties

Label 2-28e2-1.1-c5-0-10
Degree 22
Conductor 784784
Sign 11
Analytic cond. 125.740125.740
Root an. cond. 11.213411.2134
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.55·3-s − 41.6·5-s − 230.·9-s − 110.·11-s + 179.·13-s + 148.·15-s − 355.·17-s − 1.95e3·19-s − 1.54e3·23-s − 1.39e3·25-s + 1.68e3·27-s + 6.27e3·29-s − 6.00e3·31-s + 394.·33-s − 9.68e3·37-s − 637.·39-s + 1.05e4·41-s − 6.71e3·43-s + 9.59e3·45-s − 2.72e4·47-s + 1.26e3·51-s − 3.26e4·53-s + 4.61e3·55-s + 6.96e3·57-s + 492.·59-s + 4.05e4·61-s − 7.45e3·65-s + ⋯
L(s)  = 1  − 0.228·3-s − 0.744·5-s − 0.947·9-s − 0.275·11-s + 0.293·13-s + 0.170·15-s − 0.298·17-s − 1.24·19-s − 0.609·23-s − 0.445·25-s + 0.444·27-s + 1.38·29-s − 1.12·31-s + 0.0630·33-s − 1.16·37-s − 0.0671·39-s + 0.982·41-s − 0.553·43-s + 0.706·45-s − 1.79·47-s + 0.0681·51-s − 1.59·53-s + 0.205·55-s + 0.283·57-s + 0.0184·59-s + 1.39·61-s − 0.218·65-s + ⋯

Functional equation

Λ(s)=(784s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(784s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 784784    =    24⋅722^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 125.740125.740
Root analytic conductor: 11.213411.2134
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 784, ( :5/2), 1)(2,\ 784,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.55049557980.5504955798
L(12)L(\frac12) ≈\approx 0.55049557980.5504955798
L(72)L(\frac{7}{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
7 1 1
good3 1+3.55T+243T2 1 + 3.55T + 243T^{2}
5 1+41.6T+3.12e3T2 1 + 41.6T + 3.12e3T^{2}
11 1+110.T+1.61e5T2 1 + 110.T + 1.61e5T^{2}
13 1−179.T+3.71e5T2 1 - 179.T + 3.71e5T^{2}
17 1+355.T+1.41e6T2 1 + 355.T + 1.41e6T^{2}
19 1+1.95e3T+2.47e6T2 1 + 1.95e3T + 2.47e6T^{2}
23 1+1.54e3T+6.43e6T2 1 + 1.54e3T + 6.43e6T^{2}
29 1−6.27e3T+2.05e7T2 1 - 6.27e3T + 2.05e7T^{2}
31 1+6.00e3T+2.86e7T2 1 + 6.00e3T + 2.86e7T^{2}
37 1+9.68e3T+6.93e7T2 1 + 9.68e3T + 6.93e7T^{2}
41 1−1.05e4T+1.15e8T2 1 - 1.05e4T + 1.15e8T^{2}
43 1+6.71e3T+1.47e8T2 1 + 6.71e3T + 1.47e8T^{2}
47 1+2.72e4T+2.29e8T2 1 + 2.72e4T + 2.29e8T^{2}
53 1+3.26e4T+4.18e8T2 1 + 3.26e4T + 4.18e8T^{2}
59 1−492.T+7.14e8T2 1 - 492.T + 7.14e8T^{2}
61 1−4.05e4T+8.44e8T2 1 - 4.05e4T + 8.44e8T^{2}
67 1−7.68e3T+1.35e9T2 1 - 7.68e3T + 1.35e9T^{2}
71 1+7.78e4T+1.80e9T2 1 + 7.78e4T + 1.80e9T^{2}
73 1−7.39e4T+2.07e9T2 1 - 7.39e4T + 2.07e9T^{2}
79 1−4.39e4T+3.07e9T2 1 - 4.39e4T + 3.07e9T^{2}
83 1+4.11e4T+3.93e9T2 1 + 4.11e4T + 3.93e9T^{2}
89 1−6.57e4T+5.58e9T2 1 - 6.57e4T + 5.58e9T^{2}
97 1+6.85e4T+8.58e9T2 1 + 6.85e4T + 8.58e9T^{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

−9.504688903201372773616667437317, −8.405472605964590686478805477834, −8.115826042974418209714242727996, −6.85337835563673204102868182485, −6.08372543334005809965350966485, −5.06534609785773161694605554344, −4.07808164414114748464937149011, −3.10776246343891991269206119607, −1.91656849552959566734785572963, −0.32806566479988769799584845721, 0.32806566479988769799584845721, 1.91656849552959566734785572963, 3.10776246343891991269206119607, 4.07808164414114748464937149011, 5.06534609785773161694605554344, 6.08372543334005809965350966485, 6.85337835563673204102868182485, 8.115826042974418209714242727996, 8.405472605964590686478805477834, 9.504688903201372773616667437317

Graph of the ZZ-function along the critical line