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Properties

Label 2-882-1.1-c5-0-34
Degree 22
Conductor 882882
Sign −1-1
Analytic cond. 141.458141.458
Root an. cond. 11.893611.8936
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 16·4-s − 77.5·5-s − 64·8-s + 310.·10-s − 618.·11-s − 130.·13-s + 256·16-s + 263.·17-s − 91.9·19-s − 1.24e3·20-s + 2.47e3·22-s + 1.39e3·23-s + 2.89e3·25-s + 521.·26-s + 5.69e3·29-s + 6.95e3·31-s − 1.02e3·32-s − 1.05e3·34-s + 5.04e3·37-s + 367.·38-s + 4.96e3·40-s − 2.09e4·41-s − 2.13e4·43-s − 9.89e3·44-s − 5.56e3·46-s + 1.98e4·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.38·5-s − 0.353·8-s + 0.981·10-s − 1.54·11-s − 0.213·13-s + 0.250·16-s + 0.221·17-s − 0.0584·19-s − 0.694·20-s + 1.08·22-s + 0.548·23-s + 0.926·25-s + 0.151·26-s + 1.25·29-s + 1.29·31-s − 0.176·32-s − 0.156·34-s + 0.606·37-s + 0.0413·38-s + 0.490·40-s − 1.95·41-s − 1.76·43-s − 0.770·44-s − 0.387·46-s + 1.31·47-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 141.458141.458
Root analytic conductor: 11.893611.8936
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 882, ( :5/2), −1)(2,\ 882,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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+4T 1 + 4T
3 1 1
7 1 1
good5 1+77.5T+3.12e3T2 1 + 77.5T + 3.12e3T^{2}
11 1+618.T+1.61e5T2 1 + 618.T + 1.61e5T^{2}
13 1+130.T+3.71e5T2 1 + 130.T + 3.71e5T^{2}
17 1−263.T+1.41e6T2 1 - 263.T + 1.41e6T^{2}
19 1+91.9T+2.47e6T2 1 + 91.9T + 2.47e6T^{2}
23 1−1.39e3T+6.43e6T2 1 - 1.39e3T + 6.43e6T^{2}
29 1−5.69e3T+2.05e7T2 1 - 5.69e3T + 2.05e7T^{2}
31 1−6.95e3T+2.86e7T2 1 - 6.95e3T + 2.86e7T^{2}
37 1−5.04e3T+6.93e7T2 1 - 5.04e3T + 6.93e7T^{2}
41 1+2.09e4T+1.15e8T2 1 + 2.09e4T + 1.15e8T^{2}
43 1+2.13e4T+1.47e8T2 1 + 2.13e4T + 1.47e8T^{2}
47 1−1.98e4T+2.29e8T2 1 - 1.98e4T + 2.29e8T^{2}
53 1−2.75e4T+4.18e8T2 1 - 2.75e4T + 4.18e8T^{2}
59 1−4.35e4T+7.14e8T2 1 - 4.35e4T + 7.14e8T^{2}
61 1−3.39e4T+8.44e8T2 1 - 3.39e4T + 8.44e8T^{2}
67 1−1.03e4T+1.35e9T2 1 - 1.03e4T + 1.35e9T^{2}
71 1+7.69e4T+1.80e9T2 1 + 7.69e4T + 1.80e9T^{2}
73 1−1.98e3T+2.07e9T2 1 - 1.98e3T + 2.07e9T^{2}
79 1+5.38e3T+3.07e9T2 1 + 5.38e3T + 3.07e9T^{2}
83 1+6.89e4T+3.93e9T2 1 + 6.89e4T + 3.93e9T^{2}
89 1+9.04e4T+5.58e9T2 1 + 9.04e4T + 5.58e9T^{2}
97 1−1.37e5T+8.58e9T2 1 - 1.37e5T + 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

−8.554126223711287589987772092405, −8.300404256868600231453452760880, −7.43025383532482429028851935526, −6.76465136796237618407577454979, −5.40195254399666811455824754033, −4.51328996778309303297454757287, −3.30670619039744093512593138165, −2.47471824809626313682612031407, −0.875893954111922832017054695497, 0, 0.875893954111922832017054695497, 2.47471824809626313682612031407, 3.30670619039744093512593138165, 4.51328996778309303297454757287, 5.40195254399666811455824754033, 6.76465136796237618407577454979, 7.43025383532482429028851935526, 8.300404256868600231453452760880, 8.554126223711287589987772092405

Graph of the ZZ-function along the critical line