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Properties

Label 2-9702-1.1-c1-0-131
Degree 22
Conductor 97029702
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2.80·5-s − 8-s − 2.80·10-s + 11-s + 16-s − 7.96·17-s − 6.48·19-s + 2.80·20-s − 22-s + 5.28·23-s + 2.87·25-s − 5.12·29-s − 8.96·31-s − 32-s + 7.96·34-s + 10.4·37-s + 6.48·38-s − 2.80·40-s + 5.09·41-s + 11.4·43-s + 44-s − 5.28·46-s − 0.322·47-s − 2.87·50-s − 8·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.25·5-s − 0.353·8-s − 0.887·10-s + 0.301·11-s + 0.250·16-s − 1.93·17-s − 1.48·19-s + 0.627·20-s − 0.213·22-s + 1.10·23-s + 0.574·25-s − 0.952·29-s − 1.61·31-s − 0.176·32-s + 1.36·34-s + 1.72·37-s + 1.05·38-s − 0.443·40-s + 0.795·41-s + 1.74·43-s + 0.150·44-s − 0.779·46-s − 0.0471·47-s − 0.406·50-s − 1.09·53-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: −1-1
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: 11
Selberg data: (2, 9702, ( :1/2), −1)(2,\ 9702,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1−2.80T+5T2 1 - 2.80T + 5T^{2}
13 1+13T2 1 + 13T^{2}
17 1+7.96T+17T2 1 + 7.96T + 17T^{2}
19 1+6.48T+19T2 1 + 6.48T + 19T^{2}
23 1−5.28T+23T2 1 - 5.28T + 23T^{2}
29 1+5.12T+29T2 1 + 5.12T + 29T^{2}
31 1+8.96T+31T2 1 + 8.96T + 31T^{2}
37 1−10.4T+37T2 1 - 10.4T + 37T^{2}
41 1−5.09T+41T2 1 - 5.09T + 41T^{2}
43 1−11.4T+43T2 1 - 11.4T + 43T^{2}
47 1+0.322T+47T2 1 + 0.322T + 47T^{2}
53 1+8T+53T2 1 + 8T + 53T^{2}
59 1−0.871T+59T2 1 - 0.871T + 59T^{2}
61 1−2.15T+61T2 1 - 2.15T + 61T^{2}
67 1+7.09T+67T2 1 + 7.09T + 67T^{2}
71 1−7.44T+71T2 1 - 7.44T + 71T^{2}
73 1−12.5T+73T2 1 - 12.5T + 73T^{2}
79 1−6.80T+79T2 1 - 6.80T + 79T^{2}
83 1+15.0T+83T2 1 + 15.0T + 83T^{2}
89 1+1.61T+89T2 1 + 1.61T + 89T^{2}
97 1−7T+97T2 1 - 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

−7.28788622642520136789915827878, −6.60529779673015006631654424414, −6.16323737919315567129194713919, −5.50352515419386436509365950406, −4.55350280690432991381603486417, −3.85590808501459984942544076550, −2.48662868946096378495441374598, −2.23857344486020674594146839751, −1.29764244572361468263513943308, 0, 1.29764244572361468263513943308, 2.23857344486020674594146839751, 2.48662868946096378495441374598, 3.85590808501459984942544076550, 4.55350280690432991381603486417, 5.50352515419386436509365950406, 6.16323737919315567129194713919, 6.60529779673015006631654424414, 7.28788622642520136789915827878

Graph of the ZZ-function along the critical line