Properties

Label 2-768-1.1-c3-0-19
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 $0$

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: \(0\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.324097848\)
\(L(\frac12)\) \(\approx\) \(3.324097848\)
\(L(\frac{5}{2})\) 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

−9.853472022869388333724390936321, −9.228144744378559812577131908140, −8.000882744564180202558408140388, −7.68149148544036035847727738250, −6.37312873018843432793239050272, −5.42389267836855937572551007195, −4.54822785523923785456930299533, −3.21887512921733576027215084939, −2.22801397088407262310070724366, −1.07533211356721680676040244715, 1.07533211356721680676040244715, 2.22801397088407262310070724366, 3.21887512921733576027215084939, 4.54822785523923785456930299533, 5.42389267836855937572551007195, 6.37312873018843432793239050272, 7.68149148544036035847727738250, 8.000882744564180202558408140388, 9.228144744378559812577131908140, 9.853472022869388333724390936321

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