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Properties

Label 2-91e2-1.1-c1-0-316
Degree 22
Conductor 82818281
Sign −1-1
Analytic cond. 66.124166.1241
Root an. cond. 8.131678.13167
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 − 3·3-s − 4-s + 3·5-s − 3·6-s − 3·8-s + 6·9-s + 3·10-s − 3·11-s + 3·12-s − 9·15-s − 16-s − 2·17-s + 6·18-s − 19-s − 3·20-s − 3·22-s + 9·24-s + 4·25-s − 9·27-s + 7·29-s − 9·30-s + 3·31-s + 5·32-s + 9·33-s − 2·34-s − 6·36-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s − 1/2·4-s + 1.34·5-s − 1.22·6-s − 1.06·8-s + 2·9-s + 0.948·10-s − 0.904·11-s + 0.866·12-s − 2.32·15-s − 1/4·16-s − 0.485·17-s + 1.41·18-s − 0.229·19-s − 0.670·20-s − 0.639·22-s + 1.83·24-s + 4/5·25-s − 1.73·27-s + 1.29·29-s − 1.64·30-s + 0.538·31-s + 0.883·32-s + 1.56·33-s − 0.342·34-s − 36-s + ⋯

Functional equation

Λ(s)=(8281s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8281s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 82818281    =    72⋅1327^{2} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 66.124166.1241
Root analytic conductor: 8.131678.13167
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8281, ( :1/2), −1)(2,\ 8281,\ (\ :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)
bad7 1 1
13 1 1
good2 1−T+pT2 1 - T + p T^{2}
3 1+pT+pT2 1 + p T + p T^{2}
5 1−3T+pT2 1 - 3 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−7T+pT2 1 - 7 T + p T^{2}
31 1−3T+pT2 1 - 3 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1+7T+pT2 1 + 7 T + p T^{2}
47 1−T+pT2 1 - T + p T^{2}
53 1−3T+pT2 1 - 3 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+13T+pT2 1 + 13 T + p T^{2}
67 1+3T+pT2 1 + 3 T + p T^{2}
71 1−13T+pT2 1 - 13 T + p T^{2}
73 1+13T+pT2 1 + 13 T + p T^{2}
79 1+3T+pT2 1 + 3 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+5T+pT2 1 + 5 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

−7.05362532194932532109578552471, −6.30367620644786691602532205020, −5.99169125898668912910185184504, −5.41357232391496343740671029659, −4.74921498859754640487471102963, −4.45138766055725594219682573053, −3.14255396590887286261559539361, −2.21363063969416839678289895564, −1.06882102719592073190368177529, 0, 1.06882102719592073190368177529, 2.21363063969416839678289895564, 3.14255396590887286261559539361, 4.45138766055725594219682573053, 4.74921498859754640487471102963, 5.41357232391496343740671029659, 5.99169125898668912910185184504, 6.30367620644786691602532205020, 7.05362532194932532109578552471

Graph of the ZZ-function along the critical line