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Properties

Label 2-28e2-1.1-c5-0-5
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  + 4.24·3-s − 65.0·5-s − 225·9-s − 274·11-s − 746.·13-s − 276·15-s − 171.·17-s + 575.·19-s − 2.75e3·23-s + 1.10e3·25-s − 1.98e3·27-s − 3.98e3·29-s − 7.42e3·31-s − 1.16e3·33-s + 1.24e4·37-s − 3.16e3·39-s − 1.12e4·41-s − 1.70e4·43-s + 1.46e4·45-s + 1.71e4·47-s − 725.·51-s + 3.35e4·53-s + 1.78e4·55-s + 2.44e3·57-s − 5.27e4·59-s − 2.87e4·61-s + 4.85e4·65-s + ⋯
L(s)  = 1  + 0.272·3-s − 1.16·5-s − 0.925·9-s − 0.682·11-s − 1.22·13-s − 0.316·15-s − 0.143·17-s + 0.365·19-s − 1.08·23-s + 0.354·25-s − 0.524·27-s − 0.879·29-s − 1.38·31-s − 0.185·33-s + 1.49·37-s − 0.333·39-s − 1.04·41-s − 1.40·43-s + 1.07·45-s + 1.13·47-s − 0.0390·51-s + 1.64·53-s + 0.794·55-s + 0.0995·57-s − 1.97·59-s − 0.990·61-s + 1.42·65-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 0.30655438500.3065543850
L(12)L(\frac12) ≈\approx 0.30655438500.3065543850
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 1−4.24T+243T2 1 - 4.24T + 243T^{2}
5 1+65.0T+3.12e3T2 1 + 65.0T + 3.12e3T^{2}
11 1+274T+1.61e5T2 1 + 274T + 1.61e5T^{2}
13 1+746.T+3.71e5T2 1 + 746.T + 3.71e5T^{2}
17 1+171.T+1.41e6T2 1 + 171.T + 1.41e6T^{2}
19 1−575.T+2.47e6T2 1 - 575.T + 2.47e6T^{2}
23 1+2.75e3T+6.43e6T2 1 + 2.75e3T + 6.43e6T^{2}
29 1+3.98e3T+2.05e7T2 1 + 3.98e3T + 2.05e7T^{2}
31 1+7.42e3T+2.86e7T2 1 + 7.42e3T + 2.86e7T^{2}
37 1−1.24e4T+6.93e7T2 1 - 1.24e4T + 6.93e7T^{2}
41 1+1.12e4T+1.15e8T2 1 + 1.12e4T + 1.15e8T^{2}
43 1+1.70e4T+1.47e8T2 1 + 1.70e4T + 1.47e8T^{2}
47 1−1.71e4T+2.29e8T2 1 - 1.71e4T + 2.29e8T^{2}
53 1−3.35e4T+4.18e8T2 1 - 3.35e4T + 4.18e8T^{2}
59 1+5.27e4T+7.14e8T2 1 + 5.27e4T + 7.14e8T^{2}
61 1+2.87e4T+8.44e8T2 1 + 2.87e4T + 8.44e8T^{2}
67 1−1.05e4T+1.35e9T2 1 - 1.05e4T + 1.35e9T^{2}
71 1−1.05e4T+1.80e9T2 1 - 1.05e4T + 1.80e9T^{2}
73 1−2.37e4T+2.07e9T2 1 - 2.37e4T + 2.07e9T^{2}
79 1+1.03e5T+3.07e9T2 1 + 1.03e5T + 3.07e9T^{2}
83 1+1.82e4T+3.93e9T2 1 + 1.82e4T + 3.93e9T^{2}
89 1−4.73e4T+5.58e9T2 1 - 4.73e4T + 5.58e9T^{2}
97 1+4.54e4T+8.58e9T2 1 + 4.54e4T + 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

−9.470992962471521965258669858449, −8.565486551326832440270991137900, −7.72237893339769620344335054529, −7.34964237013917137550833884216, −5.93132861176516595887146190554, −5.04008038984725728382232927432, −3.98901872633266974571688997546, −3.07401814334093639232249783631, −2.08162691289654556715088404573, −0.23267906574130013221452631315, 0.23267906574130013221452631315, 2.08162691289654556715088404573, 3.07401814334093639232249783631, 3.98901872633266974571688997546, 5.04008038984725728382232927432, 5.93132861176516595887146190554, 7.34964237013917137550833884216, 7.72237893339769620344335054529, 8.565486551326832440270991137900, 9.470992962471521965258669858449

Graph of the ZZ-function along the critical line