Properties

Label 2-768-1.1-c1-0-0
Degree $2$
Conductor $768$
Sign $1$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3.46·5-s − 3.46·7-s + 9-s + 3.46·15-s + 6·17-s − 4·19-s + 3.46·21-s + 6.92·23-s + 6.99·25-s − 27-s + 3.46·29-s − 3.46·31-s + 11.9·35-s + 6.92·37-s + 6·41-s − 4·43-s − 3.46·45-s − 6.92·47-s + 4.99·49-s − 6·51-s + 3.46·53-s + 4·57-s + 12·59-s − 6.92·61-s − 3.46·63-s + 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.54·5-s − 1.30·7-s + 0.333·9-s + 0.894·15-s + 1.45·17-s − 0.917·19-s + 0.755·21-s + 1.44·23-s + 1.39·25-s − 0.192·27-s + 0.643·29-s − 0.622·31-s + 2.02·35-s + 1.13·37-s + 0.937·41-s − 0.609·43-s − 0.516·45-s − 1.01·47-s + 0.714·49-s − 0.840·51-s + 0.475·53-s + 0.529·57-s + 1.56·59-s − 0.887·61-s − 0.436·63-s + 0.488·67-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6700896478\)
\(L(\frac12)\) \(\approx\) \(0.6700896478\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.46T + 5T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 - 3.46T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 - 6.92T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 - 3.46T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 6.92T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 6.92T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.40832323178950496516690073363, −9.567824670349199995278608367568, −8.546702118283194580045683006223, −7.60853996407630508849648715281, −6.89673040000006302638164559861, −6.01226049225962676935830219852, −4.82532112152912354925956242158, −3.79580701790612401048742900272, −3.02866888416199003373007631606, −0.68078562470568089929410367530, 0.68078562470568089929410367530, 3.02866888416199003373007631606, 3.79580701790612401048742900272, 4.82532112152912354925956242158, 6.01226049225962676935830219852, 6.89673040000006302638164559861, 7.60853996407630508849648715281, 8.546702118283194580045683006223, 9.567824670349199995278608367568, 10.40832323178950496516690073363

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