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Properties

Label 2-700-1.1-c1-0-6
Degree 22
Conductor 700700
Sign 11
Analytic cond. 5.589525.58952
Root an. cond. 2.364212.36421
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 7-s + 6·9-s + 3·11-s + 13-s − 5·17-s − 8·19-s + 3·21-s + 2·23-s + 9·27-s − 29-s − 2·31-s + 9·33-s + 10·37-s + 3·39-s − 6·41-s − 4·43-s + 11·47-s + 49-s − 15·51-s + 6·53-s − 24·57-s − 10·59-s + 6·63-s − 10·67-s + 6·69-s − 10·73-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.377·7-s + 2·9-s + 0.904·11-s + 0.277·13-s − 1.21·17-s − 1.83·19-s + 0.654·21-s + 0.417·23-s + 1.73·27-s − 0.185·29-s − 0.359·31-s + 1.56·33-s + 1.64·37-s + 0.480·39-s − 0.937·41-s − 0.609·43-s + 1.60·47-s + 1/7·49-s − 2.10·51-s + 0.824·53-s − 3.17·57-s − 1.30·59-s + 0.755·63-s − 1.22·67-s + 0.722·69-s − 1.17·73-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: 11
Analytic conductor: 5.589525.58952
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 700, ( :1/2), 1)(2,\ 700,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8009323282.800932328
L(12)L(\frac12) ≈\approx 2.8009323282.800932328
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 1
5 1 1
7 1−T 1 - T
good3 1−pT+pT2 1 - p T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1+5T+pT2 1 + 5 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1+T+pT2 1 + T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−11T+pT2 1 - 11 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+7T+pT2 1 + 7 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−8T+pT2 1 - 8 T + p T^{2}
97 1−3T+pT2 1 - 3 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

−10.33350136132695345505774184140, −9.129359814341148121145330711601, −8.878316880045985327616196183506, −8.074771231686246722602580121512, −7.11680708466198240902328128179, −6.24807203668297539781117592997, −4.48739889783168415514429250228, −3.91658437283551739137960947678, −2.64505632772490851790668490036, −1.71009994790434979397954725806, 1.71009994790434979397954725806, 2.64505632772490851790668490036, 3.91658437283551739137960947678, 4.48739889783168415514429250228, 6.24807203668297539781117592997, 7.11680708466198240902328128179, 8.074771231686246722602580121512, 8.878316880045985327616196183506, 9.129359814341148121145330711601, 10.33350136132695345505774184140

Graph of the ZZ-function along the critical line