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Properties

Label 2-8820-1.1-c1-0-0
Degree 22
Conductor 88208820
Sign 11
Analytic cond. 70.428070.4280
Root an. cond. 8.392148.39214
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  − 5-s − 6·11-s − 2·13-s − 6·17-s − 8·19-s − 3·23-s + 25-s − 3·29-s − 2·31-s + 8·37-s − 3·41-s + 5·43-s − 12·53-s + 6·55-s + 61-s + 2·65-s − 7·67-s + 10·73-s − 4·79-s + 3·83-s + 6·85-s − 3·89-s + 8·95-s + 10·97-s − 3·101-s + 7·103-s + 3·107-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.80·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s − 0.625·23-s + 1/5·25-s − 0.557·29-s − 0.359·31-s + 1.31·37-s − 0.468·41-s + 0.762·43-s − 1.64·53-s + 0.809·55-s + 0.128·61-s + 0.248·65-s − 0.855·67-s + 1.17·73-s − 0.450·79-s + 0.329·83-s + 0.650·85-s − 0.317·89-s + 0.820·95-s + 1.01·97-s − 0.298·101-s + 0.689·103-s + 0.290·107-s + ⋯

Functional equation

Λ(s)=(8820s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8820s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 88208820    =    22⋅32⋅5⋅722^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}
Sign: 11
Analytic conductor: 70.428070.4280
Root analytic conductor: 8.392148.39214
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8820, ( :1/2), 1)(2,\ 8820,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.41815482440.4181548244
L(12)L(\frac12) ≈\approx 0.41815482440.4181548244
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
5 1+T 1 + T
7 1 1
good11 1+6T+pT2 1 + 6 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1+3T+pT2 1 + 3 T + p T^{2}
43 1−5T+pT2 1 - 5 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1+pT2 1 + 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 1−10T+pT2 1 - 10 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−3T+pT2 1 - 3 T + p T^{2}
89 1+3T+pT2 1 + 3 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

−7.77468326399584550521827841367, −7.20111926956575288966148646324, −6.34142829172486695403193989702, −5.77091786358778297474480067197, −4.67021280813719951130691094523, −4.53586003248224039106093854886, −3.45679418594533079489720780262, −2.43909193961244322183180369905, −2.07696906998804114155894339696, −0.28745544287901866526726671660, 0.28745544287901866526726671660, 2.07696906998804114155894339696, 2.43909193961244322183180369905, 3.45679418594533079489720780262, 4.53586003248224039106093854886, 4.67021280813719951130691094523, 5.77091786358778297474480067197, 6.34142829172486695403193989702, 7.20111926956575288966148646324, 7.77468326399584550521827841367

Graph of the ZZ-function along the critical line