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Properties

Label 2-702-1.1-c1-0-11
Degree 22
Conductor 702702
Sign −1-1
Analytic cond. 5.605495.60549
Root an. cond. 2.367592.36759
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 + 5-s − 5·7-s − 8-s − 10-s + 5·11-s − 13-s + 5·14-s + 16-s − 6·17-s − 6·19-s + 20-s − 5·22-s + 4·23-s − 4·25-s + 26-s − 5·28-s − 2·29-s + 31-s − 32-s + 6·34-s − 5·35-s − 6·37-s + 6·38-s − 40-s − 12·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.88·7-s − 0.353·8-s − 0.316·10-s + 1.50·11-s − 0.277·13-s + 1.33·14-s + 1/4·16-s − 1.45·17-s − 1.37·19-s + 0.223·20-s − 1.06·22-s + 0.834·23-s − 4/5·25-s + 0.196·26-s − 0.944·28-s − 0.371·29-s + 0.179·31-s − 0.176·32-s + 1.02·34-s − 0.845·35-s − 0.986·37-s + 0.973·38-s − 0.158·40-s − 1.87·41-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: −1-1
Analytic conductor: 5.605495.60549
Root analytic conductor: 2.367592.36759
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 702, ( :1/2), −1)(2,\ 702,\ (\ :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
13 1+T 1 + T
good5 1−T+pT2 1 - T + p T^{2}
7 1+5T+pT2 1 + 5 T + p T^{2}
11 1−5T+pT2 1 - 5 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−T+pT2 1 - T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+12T+pT2 1 + 12 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−2T+pT2 1 - 2 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+14T+pT2 1 + 14 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−2T+pT2 1 - 2 T + p T^{2}
73 1+13T+pT2 1 + 13 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+15T+pT2 1 + 15 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+7T+pT2 1 + 7 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

−9.897800505878514138826918489618, −9.064499359203391672641516805909, −8.799134072098402372739077096433, −7.10065667097095096519189382958, −6.57913304235760950474733326879, −5.99142080412013975847311847942, −4.26906324020580354961834542498, −3.19209557202869156366521081245, −1.92254093563913484185105860470, 0, 1.92254093563913484185105860470, 3.19209557202869156366521081245, 4.26906324020580354961834542498, 5.99142080412013975847311847942, 6.57913304235760950474733326879, 7.10065667097095096519189382958, 8.799134072098402372739077096433, 9.064499359203391672641516805909, 9.897800505878514138826918489618

Graph of the ZZ-function along the critical line