Properties

Label 2-768-1.1-c3-0-4
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 − 19.8·5-s + 19.8·7-s + 9·9-s + 48·11-s − 79.5·13-s + 59.6·15-s − 42·17-s − 92·19-s − 59.6·21-s − 39.7·23-s + 271·25-s − 27·27-s + 19.8·29-s − 139.·31-s − 144·33-s − 396·35-s − 198.·37-s + 238.·39-s + 6·41-s − 92·43-s − 179.·45-s + 39.7·47-s + 53·49-s + 126·51-s + 497.·53-s − 955.·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.77·5-s + 1.07·7-s + 0.333·9-s + 1.31·11-s − 1.69·13-s + 1.02·15-s − 0.599·17-s − 1.11·19-s − 0.620·21-s − 0.360·23-s + 2.16·25-s − 0.192·27-s + 0.127·29-s − 0.807·31-s − 0.759·33-s − 1.91·35-s − 0.884·37-s + 0.980·39-s + 0.0228·41-s − 0.326·43-s − 0.593·45-s + 0.123·47-s + 0.154·49-s + 0.345·51-s + 1.28·53-s − 2.34·55-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

\(L(2)\) \(\approx\) \(0.8532640670\)
\(L(\frac12)\) \(\approx\) \(0.8532640670\)
\(L(\frac{5}{2})\) 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 \( 1 + 19.8T + 125T^{2} \)
7 \( 1 - 19.8T + 343T^{2} \)
11 \( 1 - 48T + 1.33e3T^{2} \)
13 \( 1 + 79.5T + 2.19e3T^{2} \)
17 \( 1 + 42T + 4.91e3T^{2} \)
19 \( 1 + 92T + 6.85e3T^{2} \)
23 \( 1 + 39.7T + 1.21e4T^{2} \)
29 \( 1 - 19.8T + 2.43e4T^{2} \)
31 \( 1 + 139.T + 2.97e4T^{2} \)
37 \( 1 + 198.T + 5.06e4T^{2} \)
41 \( 1 - 6T + 6.89e4T^{2} \)
43 \( 1 + 92T + 7.95e4T^{2} \)
47 \( 1 - 39.7T + 1.03e5T^{2} \)
53 \( 1 - 497.T + 1.48e5T^{2} \)
59 \( 1 - 516T + 2.05e5T^{2} \)
61 \( 1 - 358.T + 2.26e5T^{2} \)
67 \( 1 - 524T + 3.00e5T^{2} \)
71 \( 1 - 994.T + 3.57e5T^{2} \)
73 \( 1 - 430T + 3.89e5T^{2} \)
79 \( 1 + 1.17e3T + 4.93e5T^{2} \)
83 \( 1 - 432T + 5.71e5T^{2} \)
89 \( 1 + 630T + 7.04e5T^{2} \)
97 \( 1 - 862T + 9.12e5T^{2} \)
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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

−10.08509825993297760030522995205, −8.856260425524114584414103424322, −8.179849312979938799124331207734, −7.25219177704208574854297326674, −6.74192302299485496203504444575, −5.18186339036491029665882696164, −4.41826592676586169956660837818, −3.79964579365562355492345370841, −2.06240059032154980978798587856, −0.52789744647342078562359004105, 0.52789744647342078562359004105, 2.06240059032154980978798587856, 3.79964579365562355492345370841, 4.41826592676586169956660837818, 5.18186339036491029665882696164, 6.74192302299485496203504444575, 7.25219177704208574854297326674, 8.179849312979938799124331207734, 8.856260425524114584414103424322, 10.08509825993297760030522995205

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