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Properties

Label 2-9702-1.1-c1-0-141
Degree 22
Conductor 97029702
Sign −1-1
Analytic cond. 77.470877.4708
Root an. cond. 8.801758.80175
Motivic weight 11
Arithmetic yes
Rational yes
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 − 4·13-s + 16-s − 3·17-s − 19-s − 22-s + 3·23-s − 5·25-s − 4·26-s + 9·29-s + 2·31-s + 32-s − 3·34-s − 7·37-s − 38-s + 6·41-s + 11·43-s − 44-s + 3·46-s + 3·47-s − 5·50-s − 4·52-s + 9·58-s − 9·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.301·11-s − 1.10·13-s + 1/4·16-s − 0.727·17-s − 0.229·19-s − 0.213·22-s + 0.625·23-s − 25-s − 0.784·26-s + 1.67·29-s + 0.359·31-s + 0.176·32-s − 0.514·34-s − 1.15·37-s − 0.162·38-s + 0.937·41-s + 1.67·43-s − 0.150·44-s + 0.442·46-s + 0.437·47-s − 0.707·50-s − 0.554·52-s + 1.18·58-s − 1.17·59-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: yes
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+pT2 1 + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−11T+pT2 1 - 11 T + p T^{2}
47 1−3T+pT2 1 - 3 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+9T+pT2 1 + 9 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+T+pT2 1 + T + p T^{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

−7.36107291702769445710144207308, −6.54821781815717450235118487434, −5.97120035176155296969755179357, −5.16567992531436349906382784247, −4.55521700469119396218663125003, −4.01662326174328454403961614589, −2.83156278620086316701868803689, −2.52019578937119137078367245192, −1.38710158730409092388824349185, 0, 1.38710158730409092388824349185, 2.52019578937119137078367245192, 2.83156278620086316701868803689, 4.01662326174328454403961614589, 4.55521700469119396218663125003, 5.16567992531436349906382784247, 5.97120035176155296969755179357, 6.54821781815717450235118487434, 7.36107291702769445710144207308

Graph of the ZZ-function along the critical line