Properties

Label 2-1323-1.1-c1-0-53
Degree $2$
Conductor $1323$
Sign $-1$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.28·2-s + 3.23·4-s − 4.23·5-s + 2.82·8-s − 9.69·10-s − 3.70·11-s + 2.28·13-s − 5·17-s − 5.45·19-s − 13.7·20-s − 8.47·22-s + 0.333·23-s + 12.9·25-s + 5.23·26-s − 7.19·29-s + 3.36·31-s − 5.65·32-s − 11.4·34-s − 4.70·37-s − 12.4·38-s − 11.9·40-s + 4.70·41-s + 2.23·43-s − 11.9·44-s + 0.763·46-s + 1.47·47-s + 29.6·50-s + ⋯
L(s)  = 1  + 1.61·2-s + 1.61·4-s − 1.89·5-s + 0.999·8-s − 3.06·10-s − 1.11·11-s + 0.634·13-s − 1.21·17-s − 1.25·19-s − 3.06·20-s − 1.80·22-s + 0.0696·23-s + 2.58·25-s + 1.02·26-s − 1.33·29-s + 0.605·31-s − 1.00·32-s − 1.96·34-s − 0.774·37-s − 2.02·38-s − 1.89·40-s + 0.735·41-s + 0.340·43-s − 1.80·44-s + 0.112·46-s + 0.214·47-s + 4.18·50-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: no
Arithmetic: yes
Character: $\chi_{1323} (1, \cdot )$
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 \( 1 - 2.28T + 2T^{2} \)
5 \( 1 + 4.23T + 5T^{2} \)
11 \( 1 + 3.70T + 11T^{2} \)
13 \( 1 - 2.28T + 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 + 5.45T + 19T^{2} \)
23 \( 1 - 0.333T + 23T^{2} \)
29 \( 1 + 7.19T + 29T^{2} \)
31 \( 1 - 3.36T + 31T^{2} \)
37 \( 1 + 4.70T + 37T^{2} \)
41 \( 1 - 4.70T + 41T^{2} \)
43 \( 1 - 2.23T + 43T^{2} \)
47 \( 1 - 1.47T + 47T^{2} \)
53 \( 1 - 3.16T + 53T^{2} \)
59 \( 1 - 3.94T + 59T^{2} \)
61 \( 1 + 3.70T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + 0.746T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 0.236T + 83T^{2} \)
89 \( 1 - 1.70T + 89T^{2} \)
97 \( 1 + 1.95T + 97T^{2} \)
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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

−8.931784417484884971309124899599, −8.236903732535624808027300572893, −7.37676777238209853721802696721, −6.68241344916331689584017786868, −5.67620303428959603353526329794, −4.63333547884182629727588088166, −4.16039531196809668671815621033, −3.37552809113310081065867172898, −2.37399114869500803124800628435, 0, 2.37399114869500803124800628435, 3.37552809113310081065867172898, 4.16039531196809668671815621033, 4.63333547884182629727588088166, 5.67620303428959603353526329794, 6.68241344916331689584017786868, 7.37676777238209853721802696721, 8.236903732535624808027300572893, 8.931784417484884971309124899599

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