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Properties

Label 2-9702-1.1-c1-0-112
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 + 0.454·5-s − 8-s − 0.454·10-s − 11-s − 0.909·13-s + 16-s + 17-s − 3.88·19-s + 0.454·20-s + 22-s − 8.33·23-s − 4.79·25-s + 0.909·26-s + 2.79·29-s + 9.58·31-s − 32-s − 34-s + 7.88·37-s + 3.88·38-s − 0.454·40-s − 1.79·41-s + 7.88·43-s − 44-s + 8.33·46-s + 0.338·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.203·5-s − 0.353·8-s − 0.143·10-s − 0.301·11-s − 0.252·13-s + 0.250·16-s + 0.242·17-s − 0.890·19-s + 0.101·20-s + 0.213·22-s − 1.73·23-s − 0.958·25-s + 0.178·26-s + 0.518·29-s + 1.72·31-s − 0.176·32-s − 0.171·34-s + 1.29·37-s + 0.629·38-s − 0.0719·40-s − 0.280·41-s + 1.20·43-s − 0.150·44-s + 1.22·46-s + 0.0493·47-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 1−0.454T+5T2 1 - 0.454T + 5T^{2}
13 1+0.909T+13T2 1 + 0.909T + 13T^{2}
17 1−T+17T2 1 - T + 17T^{2}
19 1+3.88T+19T2 1 + 3.88T + 19T^{2}
23 1+8.33T+23T2 1 + 8.33T + 23T^{2}
29 1−2.79T+29T2 1 - 2.79T + 29T^{2}
31 1−9.58T+31T2 1 - 9.58T + 31T^{2}
37 1−7.88T+37T2 1 - 7.88T + 37T^{2}
41 1+1.79T+41T2 1 + 1.79T + 41T^{2}
43 1−7.88T+43T2 1 - 7.88T + 43T^{2}
47 1−0.338T+47T2 1 - 0.338T + 47T^{2}
53 1+0.909T+53T2 1 + 0.909T + 53T^{2}
59 1−6.97T+59T2 1 - 6.97T + 59T^{2}
61 1−14.0T+61T2 1 - 14.0T + 61T^{2}
67 1+3.79T+67T2 1 + 3.79T + 67T^{2}
71 1−4.79T+71T2 1 - 4.79T + 71T^{2}
73 1+10.6T+73T2 1 + 10.6T + 73T^{2}
79 1−5.36T+79T2 1 - 5.36T + 79T^{2}
83 1−1.97T+83T2 1 - 1.97T + 83T^{2}
89 1+11.0T+89T2 1 + 11.0T + 89T^{2}
97 1+14.5T+97T2 1 + 14.5T + 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

−7.49713627177369537235960462240, −6.65249748573987834860987861573, −6.09094708357095890521726670601, −5.50455656188135923755759340654, −4.43627719664045514170067265780, −3.88914147764201965129019625467, −2.65331529557622041594585174775, −2.23684082460638736117312732227, −1.12335487363504112757685226243, 0, 1.12335487363504112757685226243, 2.23684082460638736117312732227, 2.65331529557622041594585174775, 3.88914147764201965129019625467, 4.43627719664045514170067265780, 5.50455656188135923755759340654, 6.09094708357095890521726670601, 6.65249748573987834860987861573, 7.49713627177369537235960462240

Graph of the ZZ-function along the critical line