Properties

Label 2-1323-1.1-c1-0-30
Degree $2$
Conductor $1323$
Sign $-1$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 3·5-s + 6·11-s + 4·13-s + 4·16-s − 3·17-s − 2·19-s + 6·20-s − 6·23-s + 4·25-s − 6·29-s + 4·31-s − 7·37-s + 3·41-s − 43-s − 12·44-s − 9·47-s − 8·52-s − 6·53-s − 18·55-s − 9·59-s + 10·61-s − 8·64-s − 12·65-s − 4·67-s + 6·68-s − 2·73-s + ⋯
L(s)  = 1  − 4-s − 1.34·5-s + 1.80·11-s + 1.10·13-s + 16-s − 0.727·17-s − 0.458·19-s + 1.34·20-s − 1.25·23-s + 4/5·25-s − 1.11·29-s + 0.718·31-s − 1.15·37-s + 0.468·41-s − 0.152·43-s − 1.80·44-s − 1.31·47-s − 1.10·52-s − 0.824·53-s − 2.42·55-s − 1.17·59-s + 1.28·61-s − 64-s − 1.48·65-s − 0.488·67-s + 0.727·68-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.018817159670315939703394307625, −8.510212693341528277173732259982, −7.83059910239228274735683000274, −6.73006245611635146418353618734, −5.98200354039892875610602829470, −4.62429792125361377196795523027, −3.94194391597689362310613457451, −3.55401003291548550616712145020, −1.47161332309024819880314138172, 0, 1.47161332309024819880314138172, 3.55401003291548550616712145020, 3.94194391597689362310613457451, 4.62429792125361377196795523027, 5.98200354039892875610602829470, 6.73006245611635146418353618734, 7.83059910239228274735683000274, 8.510212693341528277173732259982, 9.018817159670315939703394307625

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