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Properties

Label 2-9702-1.1-c1-0-12
Degree 22
Conductor 97029702
Sign 11
Analytic cond. 77.470877.4708
Root an. cond. 8.801758.80175
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 3.29·5-s + 8-s − 3.29·10-s + 11-s − 6.06·13-s + 16-s − 6.11·17-s − 0.0511·19-s − 3.29·20-s + 22-s + 6.75·23-s + 5.82·25-s − 6.06·26-s − 2.82·29-s − 5.87·31-s + 32-s − 6.11·34-s − 8.31·37-s − 0.0511·38-s − 3.29·40-s + 6.11·41-s + 2.90·43-s + 44-s + 6.75·46-s + 1.22·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.47·5-s + 0.353·8-s − 1.04·10-s + 0.301·11-s − 1.68·13-s + 0.250·16-s − 1.48·17-s − 0.0117·19-s − 0.735·20-s + 0.213·22-s + 1.40·23-s + 1.16·25-s − 1.19·26-s − 0.525·29-s − 1.05·31-s + 0.176·32-s − 1.04·34-s − 1.36·37-s − 0.00830·38-s − 0.520·40-s + 0.955·41-s + 0.442·43-s + 0.150·44-s + 0.996·46-s + 0.178·47-s + ⋯

Functional equation

Λ(s)=(9702s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9702s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 97029702    =    2⋅32⋅72⋅112 \cdot 3^{2} \cdot 7^{2} \cdot 11
Sign: 11
Analytic conductor: 77.470877.4708
Root analytic conductor: 8.801758.80175
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9702, ( :1/2), 1)(2,\ 9702,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4283898431.428389843
L(12)L(\frac12) ≈\approx 1.4283898431.428389843
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−T 1 - T
3 1 1
7 1 1
11 1−T 1 - T
good5 1+3.29T+5T2 1 + 3.29T + 5T^{2}
13 1+6.06T+13T2 1 + 6.06T + 13T^{2}
17 1+6.11T+17T2 1 + 6.11T + 17T^{2}
19 1+0.0511T+19T2 1 + 0.0511T + 19T^{2}
23 1−6.75T+23T2 1 - 6.75T + 23T^{2}
29 1+2.82T+29T2 1 + 2.82T + 29T^{2}
31 1+5.87T+31T2 1 + 5.87T + 31T^{2}
37 1+8.31T+37T2 1 + 8.31T + 37T^{2}
41 1−6.11T+41T2 1 - 6.11T + 41T^{2}
43 1−2.90T+43T2 1 - 2.90T + 43T^{2}
47 1−1.22T+47T2 1 - 1.22T + 47T^{2}
53 1+3.00T+53T2 1 + 3.00T + 53T^{2}
59 1−10.8T+59T2 1 - 10.8T + 59T^{2}
61 1+7.23T+61T2 1 + 7.23T + 61T^{2}
67 1+1.68T+67T2 1 + 1.68T + 67T^{2}
71 1−6.47T+71T2 1 - 6.47T + 71T^{2}
73 1−11.6T+73T2 1 - 11.6T + 73T^{2}
79 1+9.13T+79T2 1 + 9.13T + 79T^{2}
83 1−0.951T+83T2 1 - 0.951T + 83T^{2}
89 1−16.5T+89T2 1 - 16.5T + 89T^{2}
97 1−14.7T+97T2 1 - 14.7T + 97T^{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

−7.35170814516782130737453716772, −7.16798147419579201683405327998, −6.48198029136288287342807993070, −5.33985501034940280663043258970, −4.83893546751801490243629746624, −4.20242521733884771193054980402, −3.59234964998733652369681809273, −2.76327536133862309440637462533, −1.96854263231814288239933533311, −0.48397340880622314036807803054, 0.48397340880622314036807803054, 1.96854263231814288239933533311, 2.76327536133862309440637462533, 3.59234964998733652369681809273, 4.20242521733884771193054980402, 4.83893546751801490243629746624, 5.33985501034940280663043258970, 6.48198029136288287342807993070, 7.16798147419579201683405327998, 7.35170814516782130737453716772

Graph of the ZZ-function along the critical line