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Properties

Label 2-768-3.2-c0-0-2
Degree $2$
Conductor $768$
Sign $1$
Analytic cond. $0.383281$
Root an. cond. $0.619097$
Motivic weight $0$
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 + 9-s − 2·19-s + 25-s + 27-s − 2·43-s − 49-s − 2·57-s + 2·67-s − 2·73-s + 75-s + 81-s − 2·97-s + ⋯
L(s)  = 1  + 3-s + 9-s − 2·19-s + 25-s + 27-s − 2·43-s − 49-s − 2·57-s + 2·67-s − 2·73-s + 75-s + 81-s − 2·97-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $1$
Analytic conductor: \(0.383281\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{768} (257, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.308874648\)
\(L(\frac12)\) \(\approx\) \(1.308874648\)
\(L(1)\) 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 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 + T )^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T^{2} \)
67 \( ( 1 - T )^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 + T )^{2} \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 + T )^{2} \)
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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

−10.39102662824977119738323901130, −9.642217198254878916592501148423, −8.610740553163108063340259236730, −8.269999815519138953654602968242, −7.07481545445483216171663500370, −6.38352310888271867017871585322, −4.93102551732940022120334392274, −4.02671025344536349797829606424, −2.93433446474655514678138565351, −1.80861110516981843613056691839, 1.80861110516981843613056691839, 2.93433446474655514678138565351, 4.02671025344536349797829606424, 4.93102551732940022120334392274, 6.38352310888271867017871585322, 7.07481545445483216171663500370, 8.269999815519138953654602968242, 8.610740553163108063340259236730, 9.642217198254878916592501148423, 10.39102662824977119738323901130

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