Properties

Label 2-768-1.1-c3-0-44
Degree $2$
Conductor $768$
Sign $-1$
Analytic cond. $45.3134$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 2.82·5-s + 14.1·7-s + 9·9-s − 20·11-s − 39.5·13-s − 8.48·15-s − 34·17-s − 52·19-s + 42.4·21-s + 62.2·23-s − 117·25-s + 27·27-s − 200.·29-s − 110.·31-s − 60·33-s − 40.0·35-s + 271.·37-s − 118.·39-s − 26·41-s − 252·43-s − 25.4·45-s + 345.·47-s − 142.·49-s − 102·51-s + 681.·53-s + 56.5·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.252·5-s + 0.763·7-s + 0.333·9-s − 0.548·11-s − 0.844·13-s − 0.146·15-s − 0.485·17-s − 0.627·19-s + 0.440·21-s + 0.564·23-s − 0.936·25-s + 0.192·27-s − 1.28·29-s − 0.639·31-s − 0.316·33-s − 0.193·35-s + 1.20·37-s − 0.487·39-s − 0.0990·41-s − 0.893·43-s − 0.0843·45-s + 1.07·47-s − 0.416·49-s − 0.280·51-s + 1.76·53-s + 0.138·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-1$
Analytic conductor: \(45.3134\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
good5 \( 1 + 2.82T + 125T^{2} \)
7 \( 1 - 14.1T + 343T^{2} \)
11 \( 1 + 20T + 1.33e3T^{2} \)
13 \( 1 + 39.5T + 2.19e3T^{2} \)
17 \( 1 + 34T + 4.91e3T^{2} \)
19 \( 1 + 52T + 6.85e3T^{2} \)
23 \( 1 - 62.2T + 1.21e4T^{2} \)
29 \( 1 + 200.T + 2.43e4T^{2} \)
31 \( 1 + 110.T + 2.97e4T^{2} \)
37 \( 1 - 271.T + 5.06e4T^{2} \)
41 \( 1 + 26T + 6.89e4T^{2} \)
43 \( 1 + 252T + 7.95e4T^{2} \)
47 \( 1 - 345.T + 1.03e5T^{2} \)
53 \( 1 - 681.T + 1.48e5T^{2} \)
59 \( 1 + 364T + 2.05e5T^{2} \)
61 \( 1 + 735.T + 2.26e5T^{2} \)
67 \( 1 + 628T + 3.00e5T^{2} \)
71 \( 1 + 333.T + 3.57e5T^{2} \)
73 \( 1 - 338T + 3.89e5T^{2} \)
79 \( 1 + 789.T + 4.93e5T^{2} \)
83 \( 1 + 1.03e3T + 5.71e5T^{2} \)
89 \( 1 - 234T + 7.04e5T^{2} \)
97 \( 1 + 178T + 9.12e5T^{2} \)
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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

−9.395679435126117750119942833626, −8.628915338187413241312700958106, −7.71663202754461129114837029785, −7.24288418682548242680071229224, −5.87750936112110384139015108259, −4.83547845149113053283720499298, −4.00608447879461520111723008154, −2.70333168971052675208965721674, −1.74630083850338868463303793054, 0, 1.74630083850338868463303793054, 2.70333168971052675208965721674, 4.00608447879461520111723008154, 4.83547845149113053283720499298, 5.87750936112110384139015108259, 7.24288418682548242680071229224, 7.71663202754461129114837029785, 8.628915338187413241312700958106, 9.395679435126117750119942833626

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