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Properties

Label 2-84e2-1.1-c1-0-34
Degree 22
Conductor 70567056
Sign 11
Analytic cond. 56.342456.3424
Root an. cond. 7.506167.50616
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 3·11-s − 2·13-s + 3·17-s − 19-s + 3·23-s + 4·25-s + 6·29-s − 7·31-s − 37-s + 6·41-s + 4·43-s + 9·47-s − 3·53-s − 9·55-s − 9·59-s + 61-s − 6·65-s + 7·67-s + 73-s + 13·79-s − 12·83-s + 9·85-s + 15·89-s − 3·95-s + 10·97-s + 15·101-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.904·11-s − 0.554·13-s + 0.727·17-s − 0.229·19-s + 0.625·23-s + 4/5·25-s + 1.11·29-s − 1.25·31-s − 0.164·37-s + 0.937·41-s + 0.609·43-s + 1.31·47-s − 0.412·53-s − 1.21·55-s − 1.17·59-s + 0.128·61-s − 0.744·65-s + 0.855·67-s + 0.117·73-s + 1.46·79-s − 1.31·83-s + 0.976·85-s + 1.58·89-s − 0.307·95-s + 1.01·97-s + 1.49·101-s + ⋯

Functional equation

Λ(s)=(7056s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7056s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 70567056    =    24⋅32⋅722^{4} \cdot 3^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 56.342456.3424
Root analytic conductor: 7.506167.50616
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7056, ( :1/2), 1)(2,\ 7056,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5177774862.517777486
L(12)L(\frac12) ≈\approx 2.5177774862.517777486
L(32)L(\frac{3}{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 1
7 1 1
good5 1−3T+pT2 1 - 3 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+7T+pT2 1 + 7 T + p T^{2}
37 1+T+pT2 1 + T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−9T+pT2 1 - 9 T + p T^{2}
53 1+3T+pT2 1 + 3 T + p T^{2}
59 1+9T+pT2 1 + 9 T + p T^{2}
61 1−T+pT2 1 - T + p T^{2}
67 1−7T+pT2 1 - 7 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−T+pT2 1 - T + p T^{2}
79 1−13T+pT2 1 - 13 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−15T+pT2 1 - 15 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p T^{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

−7.79471766294442238351763450285, −7.33030683285885341431588133268, −6.40825653451897138853524205007, −5.78984331009086520334042076467, −5.22514419219416143835802109221, −4.57537906789274455198829131722, −3.39490893308522303308712327096, −2.56837123410425775655446467441, −1.96218931226981980488959497856, −0.804522886912069846049191220250, 0.804522886912069846049191220250, 1.96218931226981980488959497856, 2.56837123410425775655446467441, 3.39490893308522303308712327096, 4.57537906789274455198829131722, 5.22514419219416143835802109221, 5.78984331009086520334042076467, 6.40825653451897138853524205007, 7.33030683285885341431588133268, 7.79471766294442238351763450285

Graph of the ZZ-function along the critical line