Properties

Label 2-768-1.1-c1-0-14
Degree $2$
Conductor $768$
Sign $-1$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s − 2·7-s + 9-s − 4·13-s − 2·15-s − 2·17-s − 4·19-s − 2·21-s + 4·23-s − 25-s + 27-s − 6·29-s − 2·31-s + 4·35-s − 8·37-s − 4·39-s − 2·41-s − 4·43-s − 2·45-s + 12·47-s − 3·49-s − 2·51-s − 6·53-s − 4·57-s + 4·59-s − 2·63-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 1.10·13-s − 0.516·15-s − 0.485·17-s − 0.917·19-s − 0.436·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.676·35-s − 1.31·37-s − 0.640·39-s − 0.312·41-s − 0.609·43-s − 0.298·45-s + 1.75·47-s − 3/7·49-s − 0.280·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s − 0.251·63-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-1$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.774803852309850608807493449381, −9.041031444167067648097977063790, −8.195875165283705916601578714594, −7.28556468532188051006496822866, −6.69065089644886895307344321896, −5.29099993053705129646866529714, −4.18164113000798881700100095789, −3.35649033209811104812465510995, −2.19248403568752267714808595377, 0, 2.19248403568752267714808595377, 3.35649033209811104812465510995, 4.18164113000798881700100095789, 5.29099993053705129646866529714, 6.69065089644886895307344321896, 7.28556468532188051006496822866, 8.195875165283705916601578714594, 9.041031444167067648097977063790, 9.774803852309850608807493449381

Graph of the $Z$-function along the critical line