Properties

Label 2-768-1.1-c3-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 9.15·5-s − 27.4·7-s + 9·9-s − 20.5·11-s + 32.0·13-s + 27.4·15-s − 111.·17-s − 129.·19-s + 82.2·21-s − 9.16·23-s − 41.1·25-s − 27·27-s + 41.0·29-s − 187.·31-s + 61.5·33-s + 251.·35-s − 114.·37-s − 96.1·39-s − 282.·41-s + 89.3·43-s − 82.3·45-s − 54.6·47-s + 408.·49-s + 335.·51-s + 726.·53-s + 187.·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.818·5-s − 1.48·7-s + 0.333·9-s − 0.562·11-s + 0.683·13-s + 0.472·15-s − 1.59·17-s − 1.56·19-s + 0.854·21-s − 0.0830·23-s − 0.329·25-s − 0.192·27-s + 0.262·29-s − 1.08·31-s + 0.324·33-s + 1.21·35-s − 0.507·37-s − 0.394·39-s − 1.07·41-s + 0.317·43-s − 0.272·45-s − 0.169·47-s + 1.19·49-s + 0.920·51-s + 1.88·53-s + 0.460·55-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: \(0\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2791070157\)
\(L(\frac12)\) \(\approx\) \(0.2791070157\)
\(L(\frac{5}{2})\) 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 \( 1 + 9.15T + 125T^{2} \)
7 \( 1 + 27.4T + 343T^{2} \)
11 \( 1 + 20.5T + 1.33e3T^{2} \)
13 \( 1 - 32.0T + 2.19e3T^{2} \)
17 \( 1 + 111.T + 4.91e3T^{2} \)
19 \( 1 + 129.T + 6.85e3T^{2} \)
23 \( 1 + 9.16T + 1.21e4T^{2} \)
29 \( 1 - 41.0T + 2.43e4T^{2} \)
31 \( 1 + 187.T + 2.97e4T^{2} \)
37 \( 1 + 114.T + 5.06e4T^{2} \)
41 \( 1 + 282.T + 6.89e4T^{2} \)
43 \( 1 - 89.3T + 7.95e4T^{2} \)
47 \( 1 + 54.6T + 1.03e5T^{2} \)
53 \( 1 - 726.T + 1.48e5T^{2} \)
59 \( 1 - 216.T + 2.05e5T^{2} \)
61 \( 1 - 754.T + 2.26e5T^{2} \)
67 \( 1 + 379.T + 3.00e5T^{2} \)
71 \( 1 + 302.T + 3.57e5T^{2} \)
73 \( 1 - 504.T + 3.89e5T^{2} \)
79 \( 1 - 301.T + 4.93e5T^{2} \)
83 \( 1 + 599.T + 5.71e5T^{2} \)
89 \( 1 - 277.T + 7.04e5T^{2} \)
97 \( 1 + 765.T + 9.12e5T^{2} \)
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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

−10.12194775151595875939405726306, −9.000151447124826008932421458528, −8.329736859469994866281438536553, −7.03644172193551899252486464757, −6.55033827306527699334108003037, −5.60809225204336433473199396237, −4.29440452054830276877880971358, −3.61032357154715816646710094597, −2.25186256394296764612958826381, −0.28672987976436183530655822686, 0.28672987976436183530655822686, 2.25186256394296764612958826381, 3.61032357154715816646710094597, 4.29440452054830276877880971358, 5.60809225204336433473199396237, 6.55033827306527699334108003037, 7.03644172193551899252486464757, 8.329736859469994866281438536553, 9.000151447124826008932421458528, 10.12194775151595875939405726306

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