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Properties

Label 2-768-1.1-c1-0-15
Degree 22
Conductor 768768
Sign −1-1
Analytic cond. 6.132516.13251
Root an. cond. 2.476392.47639
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  + 3-s − 4·7-s + 9-s − 4·11-s − 4·13-s − 2·17-s + 4·19-s − 4·21-s − 8·23-s − 5·25-s + 27-s + 8·29-s − 4·31-s − 4·33-s + 4·37-s − 4·39-s + 6·41-s − 4·43-s − 8·47-s + 9·49-s − 2·51-s + 8·53-s + 4·57-s + 12·59-s − 12·61-s − 4·63-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 0.485·17-s + 0.917·19-s − 0.872·21-s − 1.66·23-s − 25-s + 0.192·27-s + 1.48·29-s − 0.718·31-s − 0.696·33-s + 0.657·37-s − 0.640·39-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.280·51-s + 1.09·53-s + 0.529·57-s + 1.56·59-s − 1.53·61-s − 0.503·63-s − 1.46·67-s + ⋯

Functional equation

Λ(s)=(768s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(768s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 768768    =    28⋅32^{8} \cdot 3
Sign: −1-1
Analytic conductor: 6.132516.13251
Root analytic conductor: 2.476392.47639
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 768, ( :1/2), −1)(2,\ 768,\ (\ :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 1
3 1−T 1 - T
good5 1+pT2 1 + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−4T+pT2 1 - 4 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+8T+pT2 1 + 8 T + p T^{2}
53 1−8T+pT2 1 - 8 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+12T+pT2 1 + 12 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−9.957439752177824708522207192035, −9.230110611951262626128540633521, −8.069916096323518080594124819659, −7.42165581773296115004203753691, −6.43663656032022884828664728498, −5.47500942227311527725009815404, −4.25300875683489800803673652068, −3.10975929333988114178652450581, −2.33855546999761743371608352814, 0, 2.33855546999761743371608352814, 3.10975929333988114178652450581, 4.25300875683489800803673652068, 5.47500942227311527725009815404, 6.43663656032022884828664728498, 7.42165581773296115004203753691, 8.069916096323518080594124819659, 9.230110611951262626128540633521, 9.957439752177824708522207192035

Graph of the ZZ-function along the critical line