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Properties

Label 2-9702-1.1-c1-0-115
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 + 0.414·5-s − 8-s − 0.414·10-s − 11-s − 1.17·13-s + 16-s + 2.17·17-s + 0.828·19-s + 0.414·20-s + 22-s + 3.24·23-s − 4.82·25-s + 1.17·26-s − 2.82·29-s − 6.48·31-s − 32-s − 2.17·34-s + 9.65·37-s − 0.828·38-s − 0.414·40-s − 4.65·41-s − 2.82·43-s − 44-s − 3.24·46-s + 9.24·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.185·5-s − 0.353·8-s − 0.130·10-s − 0.301·11-s − 0.324·13-s + 0.250·16-s + 0.526·17-s + 0.190·19-s + 0.0926·20-s + 0.213·22-s + 0.676·23-s − 0.965·25-s + 0.229·26-s − 0.525·29-s − 1.16·31-s − 0.176·32-s − 0.372·34-s + 1.58·37-s − 0.134·38-s − 0.0654·40-s − 0.727·41-s − 0.431·43-s − 0.150·44-s − 0.478·46-s + 1.34·47-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: −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 1−0.414T+5T2 1 - 0.414T + 5T^{2}
13 1+1.17T+13T2 1 + 1.17T + 13T^{2}
17 1−2.17T+17T2 1 - 2.17T + 17T^{2}
19 1−0.828T+19T2 1 - 0.828T + 19T^{2}
23 1−3.24T+23T2 1 - 3.24T + 23T^{2}
29 1+2.82T+29T2 1 + 2.82T + 29T^{2}
31 1+6.48T+31T2 1 + 6.48T + 31T^{2}
37 1−9.65T+37T2 1 - 9.65T + 37T^{2}
41 1+4.65T+41T2 1 + 4.65T + 41T^{2}
43 1+2.82T+43T2 1 + 2.82T + 43T^{2}
47 1−9.24T+47T2 1 - 9.24T + 47T^{2}
53 1−5.17T+53T2 1 - 5.17T + 53T^{2}
59 1−3.65T+59T2 1 - 3.65T + 59T^{2}
61 1+1.58T+61T2 1 + 1.58T + 61T^{2}
67 1+13.4T+67T2 1 + 13.4T + 67T^{2}
71 1+13.3T+71T2 1 + 13.3T + 71T^{2}
73 1+4.82T+73T2 1 + 4.82T + 73T^{2}
79 1−4.75T+79T2 1 - 4.75T + 79T^{2}
83 1−9.82T+83T2 1 - 9.82T + 83T^{2}
89 1+12.4T+89T2 1 + 12.4T + 89T^{2}
97 1+10.1T+97T2 1 + 10.1T + 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

−7.51398390608683983306335141196, −6.86107081388963457720767032075, −5.89118354338543739496892010401, −5.54178261197725231872917358725, −4.58348046191247645198656524527, −3.69291415934226054190917048706, −2.86499944919132638862752954716, −2.07805540155464135333838298770, −1.16623272416659810865035025169, 0, 1.16623272416659810865035025169, 2.07805540155464135333838298770, 2.86499944919132638862752954716, 3.69291415934226054190917048706, 4.58348046191247645198656524527, 5.54178261197725231872917358725, 5.89118354338543739496892010401, 6.86107081388963457720767032075, 7.51398390608683983306335141196

Graph of the ZZ-function along the critical line