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Properties

Label 2-882-1.1-c5-0-54
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 − 103.·5-s + 64·8-s − 413.·10-s − 240.·11-s + 805.·13-s + 256·16-s + 1.29e3·17-s − 275.·19-s − 1.65e3·20-s − 960.·22-s − 3.79e3·23-s + 7.58e3·25-s + 3.22e3·26-s − 1.22e3·29-s + 5.62e3·31-s + 1.02e3·32-s + 5.17e3·34-s − 9.07e3·37-s − 1.10e3·38-s − 6.62e3·40-s + 1.82e4·41-s − 1.17e4·43-s − 3.84e3·44-s − 1.51e4·46-s + 2.30e4·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.85·5-s + 0.353·8-s − 1.30·10-s − 0.598·11-s + 1.32·13-s + 0.250·16-s + 1.08·17-s − 0.175·19-s − 0.925·20-s − 0.423·22-s − 1.49·23-s + 2.42·25-s + 0.934·26-s − 0.271·29-s + 1.05·31-s + 0.176·32-s + 0.767·34-s − 1.09·37-s − 0.123·38-s − 0.654·40-s + 1.69·41-s − 0.965·43-s − 0.299·44-s − 1.05·46-s + 1.52·47-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+103.T+3.12e3T2 1 + 103.T + 3.12e3T^{2}
11 1+240.T+1.61e5T2 1 + 240.T + 1.61e5T^{2}
13 1−805.T+3.71e5T2 1 - 805.T + 3.71e5T^{2}
17 1−1.29e3T+1.41e6T2 1 - 1.29e3T + 1.41e6T^{2}
19 1+275.T+2.47e6T2 1 + 275.T + 2.47e6T^{2}
23 1+3.79e3T+6.43e6T2 1 + 3.79e3T + 6.43e6T^{2}
29 1+1.22e3T+2.05e7T2 1 + 1.22e3T + 2.05e7T^{2}
31 1−5.62e3T+2.86e7T2 1 - 5.62e3T + 2.86e7T^{2}
37 1+9.07e3T+6.93e7T2 1 + 9.07e3T + 6.93e7T^{2}
41 1−1.82e4T+1.15e8T2 1 - 1.82e4T + 1.15e8T^{2}
43 1+1.17e4T+1.47e8T2 1 + 1.17e4T + 1.47e8T^{2}
47 1−2.30e4T+2.29e8T2 1 - 2.30e4T + 2.29e8T^{2}
53 1+1.76e4T+4.18e8T2 1 + 1.76e4T + 4.18e8T^{2}
59 1+1.83e4T+7.14e8T2 1 + 1.83e4T + 7.14e8T^{2}
61 1−1.13e4T+8.44e8T2 1 - 1.13e4T + 8.44e8T^{2}
67 1−3.60e4T+1.35e9T2 1 - 3.60e4T + 1.35e9T^{2}
71 1−6.34e4T+1.80e9T2 1 - 6.34e4T + 1.80e9T^{2}
73 1+5.29e4T+2.07e9T2 1 + 5.29e4T + 2.07e9T^{2}
79 1+4.85e4T+3.07e9T2 1 + 4.85e4T + 3.07e9T^{2}
83 1+1.13e5T+3.93e9T2 1 + 1.13e5T + 3.93e9T^{2}
89 1+1.08e5T+5.58e9T2 1 + 1.08e5T + 5.58e9T^{2}
97 1−9.96e4T+8.58e9T2 1 - 9.96e4T + 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

−8.590615522744635089646514918710, −8.016107556012015818949064595035, −7.38961933418289585681318955410, −6.31815670830155299166156541536, −5.36774037980630252284283547740, −4.20775747082601207721577958367, −3.74490234916446448722665794607, −2.81355463206960669057603142382, −1.19102157271984553626278081789, 0, 1.19102157271984553626278081789, 2.81355463206960669057603142382, 3.74490234916446448722665794607, 4.20775747082601207721577958367, 5.36774037980630252284283547740, 6.31815670830155299166156541536, 7.38961933418289585681318955410, 8.016107556012015818949064595035, 8.590615522744635089646514918710

Graph of the ZZ-function along the critical line