Properties

Label 2-768-1.1-c1-0-6
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 $0$

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

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

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: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.172656166\)
\(L(\frac12)\) \(\approx\) \(2.172656166\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−10.46186321886751362762916978052, −9.298383051990469888672283644214, −8.520382921239941720035881428633, −7.84363031960006724677694678337, −7.15944714304425705232693435926, −5.67206581946498649675335111392, −4.95203593728251161575038294540, −3.84720863319685552465873529850, −2.59900951660796699369904554010, −1.39752451983425368394719286578, 1.39752451983425368394719286578, 2.59900951660796699369904554010, 3.84720863319685552465873529850, 4.95203593728251161575038294540, 5.67206581946498649675335111392, 7.15944714304425705232693435926, 7.84363031960006724677694678337, 8.520382921239941720035881428633, 9.298383051990469888672283644214, 10.46186321886751362762916978052

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