Properties

Label 2-768-1.1-c3-0-26
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 − 10.3·5-s − 3.46·7-s + 9·9-s + 55.4·13-s + 31.1·15-s − 90·17-s + 116·19-s + 10.3·21-s − 103.·23-s − 17·25-s − 27·27-s + 259.·29-s + 301.·31-s + 36·35-s − 34.6·37-s − 166.·39-s + 54·41-s + 20·43-s − 93.5·45-s − 394.·47-s − 331·49-s + 270·51-s − 488.·53-s − 348·57-s + 324·59-s − 575.·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.929·5-s − 0.187·7-s + 0.333·9-s + 1.18·13-s + 0.536·15-s − 1.28·17-s + 1.40·19-s + 0.107·21-s − 0.942·23-s − 0.136·25-s − 0.192·27-s + 1.66·29-s + 1.74·31-s + 0.173·35-s − 0.153·37-s − 0.682·39-s + 0.205·41-s + 0.0709·43-s − 0.309·45-s − 1.22·47-s − 0.965·49-s + 0.741·51-s − 1.26·53-s − 0.808·57-s + 0.714·59-s − 1.20·61-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 \( 1 + 10.3T + 125T^{2} \)
7 \( 1 + 3.46T + 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 55.4T + 2.19e3T^{2} \)
17 \( 1 + 90T + 4.91e3T^{2} \)
19 \( 1 - 116T + 6.85e3T^{2} \)
23 \( 1 + 103.T + 1.21e4T^{2} \)
29 \( 1 - 259.T + 2.43e4T^{2} \)
31 \( 1 - 301.T + 2.97e4T^{2} \)
37 \( 1 + 34.6T + 5.06e4T^{2} \)
41 \( 1 - 54T + 6.89e4T^{2} \)
43 \( 1 - 20T + 7.95e4T^{2} \)
47 \( 1 + 394.T + 1.03e5T^{2} \)
53 \( 1 + 488.T + 1.48e5T^{2} \)
59 \( 1 - 324T + 2.05e5T^{2} \)
61 \( 1 + 575.T + 2.26e5T^{2} \)
67 \( 1 + 116T + 3.00e5T^{2} \)
71 \( 1 - 1.10e3T + 3.57e5T^{2} \)
73 \( 1 + 1.10e3T + 3.89e5T^{2} \)
79 \( 1 - 148.T + 4.93e5T^{2} \)
83 \( 1 + 1.15e3T + 5.71e5T^{2} \)
89 \( 1 + 918T + 7.04e5T^{2} \)
97 \( 1 - 190T + 9.12e5T^{2} \)
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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

−9.608973637026612454610298084311, −8.460696333250665779317620920861, −7.88793090850533178523831913072, −6.71342733581175856866836766737, −6.13595186029342502172068421409, −4.84854109345611160543512489700, −4.05900479143116480280775166821, −2.98253769503651987511559744771, −1.25966626201570182391997511822, 0, 1.25966626201570182391997511822, 2.98253769503651987511559744771, 4.05900479143116480280775166821, 4.84854109345611160543512489700, 6.13595186029342502172068421409, 6.71342733581175856866836766737, 7.88793090850533178523831913072, 8.460696333250665779317620920861, 9.608973637026612454610298084311

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