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Properties

Label 2-9702-1.1-c1-0-122
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 − 8-s + 11-s + 4.24·13-s + 16-s + 2.82·17-s − 4.24·19-s − 22-s − 6·23-s − 5·25-s − 4.24·26-s + 4·29-s − 7.07·31-s − 32-s − 2.82·34-s + 2·37-s + 4.24·38-s − 2.82·41-s + 10·43-s + 44-s + 6·46-s + 12.7·47-s + 5·50-s + 4.24·52-s − 2·53-s − 4·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.353·8-s + 0.301·11-s + 1.17·13-s + 0.250·16-s + 0.685·17-s − 0.973·19-s − 0.213·22-s − 1.25·23-s − 25-s − 0.832·26-s + 0.742·29-s − 1.27·31-s − 0.176·32-s − 0.485·34-s + 0.328·37-s + 0.688·38-s − 0.441·41-s + 1.52·43-s + 0.150·44-s + 0.884·46-s + 1.85·47-s + 0.707·50-s + 0.588·52-s − 0.274·53-s − 0.525·58-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+5T2 1 + 5T^{2}
13 1−4.24T+13T2 1 - 4.24T + 13T^{2}
17 1−2.82T+17T2 1 - 2.82T + 17T^{2}
19 1+4.24T+19T2 1 + 4.24T + 19T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
29 1−4T+29T2 1 - 4T + 29T^{2}
31 1+7.07T+31T2 1 + 7.07T + 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+2.82T+41T2 1 + 2.82T + 41T^{2}
43 1−10T+43T2 1 - 10T + 43T^{2}
47 1−12.7T+47T2 1 - 12.7T + 47T^{2}
53 1+2T+53T2 1 + 2T + 53T^{2}
59 1+11.3T+59T2 1 + 11.3T + 59T^{2}
61 1+9.89T+61T2 1 + 9.89T + 61T^{2}
67 1−8T+67T2 1 - 8T + 67T^{2}
71 1+16T+71T2 1 + 16T + 71T^{2}
73 1+8.48T+73T2 1 + 8.48T + 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1−12.7T+83T2 1 - 12.7T + 83T^{2}
89 1+7.07T+89T2 1 + 7.07T + 89T^{2}
97 1+7.07T+97T2 1 + 7.07T + 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

−7.61306098166435554070562429733, −6.65384527303565678162733941289, −5.96843709024867243502708045318, −5.67611566136597490879348210320, −4.30847609020242236869298969714, −3.87591841174532161644364408016, −2.91948815150303705673915894821, −1.96006000917457453118379825711, −1.22292499567795485421307076375, 0, 1.22292499567795485421307076375, 1.96006000917457453118379825711, 2.91948815150303705673915894821, 3.87591841174532161644364408016, 4.30847609020242236869298969714, 5.67611566136597490879348210320, 5.96843709024867243502708045318, 6.65384527303565678162733941289, 7.61306098166435554070562429733

Graph of the ZZ-function along the critical line