Properties

Label 2-1323-1.1-c3-0-131
Degree $2$
Conductor $1323$
Sign $-1$
Analytic cond. $78.0595$
Root an. cond. $8.83513$
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  + 1.57·2-s − 5.53·4-s + 6.97·5-s − 21.2·8-s + 10.9·10-s − 32.5·11-s + 19.3·13-s + 10.8·16-s + 22.2·17-s + 155.·19-s − 38.5·20-s − 51.1·22-s − 76.9·23-s − 76.3·25-s + 30.3·26-s + 122.·29-s − 164.·31-s + 187.·32-s + 34.9·34-s + 66.2·37-s + 243.·38-s − 148.·40-s − 231.·41-s − 210.·43-s + 180.·44-s − 120.·46-s + 206.·47-s + ⋯
L(s)  = 1  + 0.555·2-s − 0.691·4-s + 0.623·5-s − 0.939·8-s + 0.346·10-s − 0.892·11-s + 0.412·13-s + 0.169·16-s + 0.317·17-s + 1.87·19-s − 0.431·20-s − 0.495·22-s − 0.697·23-s − 0.611·25-s + 0.229·26-s + 0.781·29-s − 0.954·31-s + 1.03·32-s + 0.176·34-s + 0.294·37-s + 1.04·38-s − 0.585·40-s − 0.883·41-s − 0.746·43-s + 0.617·44-s − 0.387·46-s + 0.640·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(78.0595\)
Root analytic conductor: \(8.83513\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1323,\ (\ :3/2),\ -1)\)

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)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - 1.57T + 8T^{2} \)
5 \( 1 - 6.97T + 125T^{2} \)
11 \( 1 + 32.5T + 1.33e3T^{2} \)
13 \( 1 - 19.3T + 2.19e3T^{2} \)
17 \( 1 - 22.2T + 4.91e3T^{2} \)
19 \( 1 - 155.T + 6.85e3T^{2} \)
23 \( 1 + 76.9T + 1.21e4T^{2} \)
29 \( 1 - 122.T + 2.43e4T^{2} \)
31 \( 1 + 164.T + 2.97e4T^{2} \)
37 \( 1 - 66.2T + 5.06e4T^{2} \)
41 \( 1 + 231.T + 6.89e4T^{2} \)
43 \( 1 + 210.T + 7.95e4T^{2} \)
47 \( 1 - 206.T + 1.03e5T^{2} \)
53 \( 1 + 419.T + 1.48e5T^{2} \)
59 \( 1 - 300.T + 2.05e5T^{2} \)
61 \( 1 + 24.2T + 2.26e5T^{2} \)
67 \( 1 + 274.T + 3.00e5T^{2} \)
71 \( 1 + 336.T + 3.57e5T^{2} \)
73 \( 1 + 693.T + 3.89e5T^{2} \)
79 \( 1 + 584.T + 4.93e5T^{2} \)
83 \( 1 + 1.20e3T + 5.71e5T^{2} \)
89 \( 1 + 1.25e3T + 7.04e5T^{2} \)
97 \( 1 + 1.08e3T + 9.12e5T^{2} \)
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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

−8.888976124718849821809707576307, −8.094755223078975328875814846104, −7.23233142056895624627112305396, −5.94463608891025746006222168207, −5.52773173732753900659075847118, −4.70876439749632924683662116374, −3.60503741400000970699331310004, −2.80216602775170896299156825174, −1.39054773791052678107152119524, 0, 1.39054773791052678107152119524, 2.80216602775170896299156825174, 3.60503741400000970699331310004, 4.70876439749632924683662116374, 5.52773173732753900659075847118, 5.94463608891025746006222168207, 7.23233142056895624627112305396, 8.094755223078975328875814846104, 8.888976124718849821809707576307

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