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Properties

Label 2-9702-1.1-c1-0-102
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 − 1.41·5-s − 8-s + 1.41·10-s − 11-s + 1.41·13-s + 16-s + 1.41·17-s − 2.82·19-s − 1.41·20-s + 22-s + 4·23-s − 2.99·25-s − 1.41·26-s + 6·29-s − 2.82·31-s − 32-s − 1.41·34-s − 8·37-s + 2.82·38-s + 1.41·40-s + 7.07·41-s − 8·43-s − 44-s − 4·46-s + 8.48·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.632·5-s − 0.353·8-s + 0.447·10-s − 0.301·11-s + 0.392·13-s + 0.250·16-s + 0.342·17-s − 0.648·19-s − 0.316·20-s + 0.213·22-s + 0.834·23-s − 0.599·25-s − 0.277·26-s + 1.11·29-s − 0.508·31-s − 0.176·32-s − 0.242·34-s − 1.31·37-s + 0.458·38-s + 0.223·40-s + 1.10·41-s − 1.21·43-s − 0.150·44-s − 0.589·46-s + 1.23·47-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+1.41T+5T2 1 + 1.41T + 5T^{2}
13 1−1.41T+13T2 1 - 1.41T + 13T^{2}
17 1−1.41T+17T2 1 - 1.41T + 17T^{2}
19 1+2.82T+19T2 1 + 2.82T + 19T^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1+2.82T+31T2 1 + 2.82T + 31T^{2}
37 1+8T+37T2 1 + 8T + 37T^{2}
41 1−7.07T+41T2 1 - 7.07T + 41T^{2}
43 1+8T+43T2 1 + 8T + 43T^{2}
47 1−8.48T+47T2 1 - 8.48T + 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+7.07T+61T2 1 + 7.07T + 61T^{2}
67 1+8T+67T2 1 + 8T + 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1−1.41T+73T2 1 - 1.41T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−2.82T+83T2 1 - 2.82T + 83T^{2}
89 1+9.89T+89T2 1 + 9.89T + 89T^{2}
97 1−15.5T+97T2 1 - 15.5T + 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

−7.44101026525792074041023290162, −6.83241933588875437708540172818, −6.11686917129522057351557448824, −5.35074155741522510869068543679, −4.52228542407737730630259468249, −3.69952845422182766797975115257, −2.99428346365492788390692356300, −2.06572600660909889910839720824, −1.06399021800454625787883543368, 0, 1.06399021800454625787883543368, 2.06572600660909889910839720824, 2.99428346365492788390692356300, 3.69952845422182766797975115257, 4.52228542407737730630259468249, 5.35074155741522510869068543679, 6.11686917129522057351557448824, 6.83241933588875437708540172818, 7.44101026525792074041023290162

Graph of the ZZ-function along the critical line