Properties

Label 2-768-1.1-c3-0-34
Degree $2$
Conductor $768$
Sign $-1$
Analytic cond. $45.3134$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 8·5-s − 12·7-s + 9·9-s + 12·11-s − 20·13-s − 24·15-s + 62·17-s − 108·19-s + 36·21-s + 72·23-s − 61·25-s − 27·27-s + 128·29-s − 204·31-s − 36·33-s − 96·35-s + 228·37-s + 60·39-s + 22·41-s + 204·43-s + 72·45-s − 600·47-s − 199·49-s − 186·51-s − 256·53-s + 96·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.715·5-s − 0.647·7-s + 1/3·9-s + 0.328·11-s − 0.426·13-s − 0.413·15-s + 0.884·17-s − 1.30·19-s + 0.374·21-s + 0.652·23-s − 0.487·25-s − 0.192·27-s + 0.819·29-s − 1.18·31-s − 0.189·33-s − 0.463·35-s + 1.01·37-s + 0.246·39-s + 0.0838·41-s + 0.723·43-s + 0.238·45-s − 1.86·47-s − 0.580·49-s − 0.510·51-s − 0.663·53-s + 0.235·55-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: yes
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 \( 1 + p T \)
good5 \( 1 - 8 T + p^{3} T^{2} \)
7 \( 1 + 12 T + p^{3} T^{2} \)
11 \( 1 - 12 T + p^{3} T^{2} \)
13 \( 1 + 20 T + p^{3} T^{2} \)
17 \( 1 - 62 T + p^{3} T^{2} \)
19 \( 1 + 108 T + p^{3} T^{2} \)
23 \( 1 - 72 T + p^{3} T^{2} \)
29 \( 1 - 128 T + p^{3} T^{2} \)
31 \( 1 + 204 T + p^{3} T^{2} \)
37 \( 1 - 228 T + p^{3} T^{2} \)
41 \( 1 - 22 T + p^{3} T^{2} \)
43 \( 1 - 204 T + p^{3} T^{2} \)
47 \( 1 + 600 T + p^{3} T^{2} \)
53 \( 1 + 256 T + p^{3} T^{2} \)
59 \( 1 - 828 T + p^{3} T^{2} \)
61 \( 1 - 84 T + p^{3} T^{2} \)
67 \( 1 + 348 T + p^{3} T^{2} \)
71 \( 1 + 456 T + p^{3} T^{2} \)
73 \( 1 + 822 T + p^{3} T^{2} \)
79 \( 1 + 1356 T + p^{3} T^{2} \)
83 \( 1 + 108 T + p^{3} T^{2} \)
89 \( 1 - 938 T + p^{3} T^{2} \)
97 \( 1 - 1278 T + p^{3} T^{2} \)
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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

−9.709089017802077552928223094612, −8.816524948423033486040604656330, −7.66119961588397644449976555354, −6.64197027930972812527204934561, −6.04091871751163946138532994684, −5.13294341286929820559756673521, −4.02181567853693452308754709488, −2.75242184242609664235087524851, −1.46450932401742510885149468225, 0, 1.46450932401742510885149468225, 2.75242184242609664235087524851, 4.02181567853693452308754709488, 5.13294341286929820559756673521, 6.04091871751163946138532994684, 6.64197027930972812527204934561, 7.66119961588397644449976555354, 8.816524948423033486040604656330, 9.709089017802077552928223094612

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