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Properties

Label 2-9702-1.1-c1-0-78
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 + 2.27·5-s + 8-s + 2.27·10-s + 11-s − 4.63·13-s + 16-s − 0.554·17-s + 4.07·19-s + 2.27·20-s + 22-s + 6.93·23-s + 0.171·25-s − 4.63·26-s + 2.82·29-s + 7.68·31-s + 32-s − 0.554·34-s + 10.8·37-s + 4.07·38-s + 2.27·40-s + 0.554·41-s − 8.59·43-s + 44-s + 6.93·46-s − 10.9·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.01·5-s + 0.353·8-s + 0.719·10-s + 0.301·11-s − 1.28·13-s + 0.250·16-s − 0.134·17-s + 0.935·19-s + 0.508·20-s + 0.213·22-s + 1.44·23-s + 0.0343·25-s − 0.908·26-s + 0.525·29-s + 1.38·31-s + 0.176·32-s − 0.0950·34-s + 1.78·37-s + 0.661·38-s + 0.359·40-s + 0.0865·41-s − 1.31·43-s + 0.150·44-s + 1.02·46-s − 1.59·47-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 4.4458919014.445891901
L(12)L(\frac12) ≈\approx 4.4458919014.445891901
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 1−2.27T+5T2 1 - 2.27T + 5T^{2}
13 1+4.63T+13T2 1 + 4.63T + 13T^{2}
17 1+0.554T+17T2 1 + 0.554T + 17T^{2}
19 1−4.07T+19T2 1 - 4.07T + 19T^{2}
23 1−6.93T+23T2 1 - 6.93T + 23T^{2}
29 1−2.82T+29T2 1 - 2.82T + 29T^{2}
31 1−7.68T+31T2 1 - 7.68T + 31T^{2}
37 1−10.8T+37T2 1 - 10.8T + 37T^{2}
41 1−0.554T+41T2 1 - 0.554T + 41T^{2}
43 1+8.59T+43T2 1 + 8.59T + 43T^{2}
47 1+10.9T+47T2 1 + 10.9T + 47T^{2}
53 1−0.440T+53T2 1 - 0.440T + 53T^{2}
59 1+5.17T+59T2 1 + 5.17T + 59T^{2}
61 1−2.19T+61T2 1 - 2.19T + 61T^{2}
67 1+11.1T+67T2 1 + 11.1T + 67T^{2}
71 1+3.60T+71T2 1 + 3.60T + 71T^{2}
73 1−14.3T+73T2 1 - 14.3T + 73T^{2}
79 1−15.7T+79T2 1 - 15.7T + 79T^{2}
83 1−6.51T+83T2 1 - 6.51T + 83T^{2}
89 1−4.23T+89T2 1 - 4.23T + 89T^{2}
97 1−3.84T+97T2 1 - 3.84T + 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

−7.59707159655992327236814887599, −6.67043884478156802575683240672, −6.43013654149151164692283057770, −5.48552228788439949160136406502, −4.94508857249820285055012393814, −4.45145323258622128395809210767, −3.23325929674411569881889109414, −2.73387894166343282327866351614, −1.90018564425422211803507168312, −0.925643581893000526428842730780, 0.925643581893000526428842730780, 1.90018564425422211803507168312, 2.73387894166343282327866351614, 3.23325929674411569881889109414, 4.45145323258622128395809210767, 4.94508857249820285055012393814, 5.48552228788439949160136406502, 6.43013654149151164692283057770, 6.67043884478156802575683240672, 7.59707159655992327236814887599

Graph of the ZZ-function along the critical line