Properties

Label 2-768-1.1-c3-0-41
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 $1$

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: \(1\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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

−9.593330165901153952810043278590, −8.590788082878218212723863092108, −7.84658083261436586516033190534, −6.73429815944415528067111399346, −5.93016017244913180203494266778, −4.86426477951626204554159407993, −4.43426542191552671661267341898, −2.41312144897859229375571832259, −1.69030753949563596705831871340, 0, 1.69030753949563596705831871340, 2.41312144897859229375571832259, 4.43426542191552671661267341898, 4.86426477951626204554159407993, 5.93016017244913180203494266778, 6.73429815944415528067111399346, 7.84658083261436586516033190534, 8.590788082878218212723863092108, 9.593330165901153952810043278590

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