Properties

Label 2-768-1.1-c3-0-2
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 − 3.46·5-s − 24.2·7-s + 9·9-s − 48·11-s − 41.5·13-s + 10.3·15-s + 54·17-s + 4·19-s + 72.7·21-s − 173.·23-s − 113·25-s − 27·27-s − 162.·29-s + 58.8·31-s + 144·33-s + 84·35-s + 325.·37-s + 124.·39-s + 294·41-s − 188·43-s − 31.1·45-s + 505.·47-s + 245·49-s − 162·51-s − 744.·53-s + 166.·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.309·5-s − 1.30·7-s + 0.333·9-s − 1.31·11-s − 0.886·13-s + 0.178·15-s + 0.770·17-s + 0.0482·19-s + 0.755·21-s − 1.57·23-s − 0.904·25-s − 0.192·27-s − 1.04·29-s + 0.341·31-s + 0.759·33-s + 0.405·35-s + 1.44·37-s + 0.512·39-s + 1.11·41-s − 0.666·43-s − 0.103·45-s + 1.56·47-s + 0.714·49-s − 0.444·51-s − 1.93·53-s + 0.407·55-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

\(L(2)\) \(\approx\) \(0.4696134449\)
\(L(\frac12)\) \(\approx\) \(0.4696134449\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.46T + 125T^{2} \)
7 \( 1 + 24.2T + 343T^{2} \)
11 \( 1 + 48T + 1.33e3T^{2} \)
13 \( 1 + 41.5T + 2.19e3T^{2} \)
17 \( 1 - 54T + 4.91e3T^{2} \)
19 \( 1 - 4T + 6.85e3T^{2} \)
23 \( 1 + 173.T + 1.21e4T^{2} \)
29 \( 1 + 162.T + 2.43e4T^{2} \)
31 \( 1 - 58.8T + 2.97e4T^{2} \)
37 \( 1 - 325.T + 5.06e4T^{2} \)
41 \( 1 - 294T + 6.89e4T^{2} \)
43 \( 1 + 188T + 7.95e4T^{2} \)
47 \( 1 - 505.T + 1.03e5T^{2} \)
53 \( 1 + 744.T + 1.48e5T^{2} \)
59 \( 1 + 252T + 2.05e5T^{2} \)
61 \( 1 - 90.0T + 2.26e5T^{2} \)
67 \( 1 + 628T + 3.00e5T^{2} \)
71 \( 1 - 6.92T + 3.57e5T^{2} \)
73 \( 1 - 1.00e3T + 3.89e5T^{2} \)
79 \( 1 + 1.34e3T + 4.93e5T^{2} \)
83 \( 1 - 720T + 5.71e5T^{2} \)
89 \( 1 - 1.48e3T + 7.04e5T^{2} \)
97 \( 1 - 1.82e3T + 9.12e5T^{2} \)
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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

−9.914727856306761399836281529361, −9.431466408089082362370388150270, −7.84640976633026993037644784671, −7.52829506010071625537216753328, −6.21292005604433094456829808534, −5.66868319501735855968916605829, −4.50453743016072788566029207783, −3.39948599498746099996945883740, −2.30133441456184581917067709897, −0.37220561486953340513308425231, 0.37220561486953340513308425231, 2.30133441456184581917067709897, 3.39948599498746099996945883740, 4.50453743016072788566029207783, 5.66868319501735855968916605829, 6.21292005604433094456829808534, 7.52829506010071625537216753328, 7.84640976633026993037644784671, 9.431466408089082362370388150270, 9.914727856306761399836281529361

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