Properties

Label 2-1323-1.1-c1-0-36
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 2·13-s + 4·16-s + 7·19-s − 5·25-s − 11·31-s − 10·37-s − 13·43-s + 4·52-s + 13·61-s − 8·64-s − 16·67-s + 7·73-s − 14·76-s − 4·79-s − 5·97-s + 10·100-s − 20·103-s − 19·109-s + ⋯
L(s)  = 1  − 4-s − 0.554·13-s + 16-s + 1.60·19-s − 25-s − 1.97·31-s − 1.64·37-s − 1.98·43-s + 0.554·52-s + 1.66·61-s − 64-s − 1.95·67-s + 0.819·73-s − 1.60·76-s − 0.450·79-s − 0.507·97-s + 100-s − 1.97·103-s − 1.81·109-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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−9.348577680569783908428726498396, −8.477561354971015169967731719682, −7.66380021222276853297564329904, −6.90688998756717719245467536942, −5.50635879764846365024871064576, −5.18223370651071093866018001979, −3.97618514265534826427174210608, −3.21180480093563962996990541956, −1.62851546714056177560860339476, 0, 1.62851546714056177560860339476, 3.21180480093563962996990541956, 3.97618514265534826427174210608, 5.18223370651071093866018001979, 5.50635879764846365024871064576, 6.90688998756717719245467536942, 7.66380021222276853297564329904, 8.477561354971015169967731719682, 9.348577680569783908428726498396

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