Properties

Label 2-1323-1.1-c1-0-32
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  − 2·2-s + 2·4-s + 5-s − 2·10-s − 4·11-s + 2·13-s − 4·16-s − 3·17-s + 8·19-s + 2·20-s + 8·22-s − 6·23-s − 4·25-s − 4·26-s − 4·29-s − 6·31-s + 8·32-s + 6·34-s − 3·37-s − 16·38-s − 41-s + 11·43-s − 8·44-s + 12·46-s − 9·47-s + 8·50-s + 4·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.447·5-s − 0.632·10-s − 1.20·11-s + 0.554·13-s − 16-s − 0.727·17-s + 1.83·19-s + 0.447·20-s + 1.70·22-s − 1.25·23-s − 4/5·25-s − 0.784·26-s − 0.742·29-s − 1.07·31-s + 1.41·32-s + 1.02·34-s − 0.493·37-s − 2.59·38-s − 0.156·41-s + 1.67·43-s − 1.20·44-s + 1.76·46-s − 1.31·47-s + 1.13·50-s + 0.554·52-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 \( 1 + p T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−9.310424015638182849224826873304, −8.497771804391192154093576901064, −7.69931536429066472922525605470, −7.21149427744505179683222147621, −5.96598102544836283817916841913, −5.23983624007258288531004002941, −3.90524998328143242553926635429, −2.52420740459652612310538151640, −1.53430709834964791772483315213, 0, 1.53430709834964791772483315213, 2.52420740459652612310538151640, 3.90524998328143242553926635429, 5.23983624007258288531004002941, 5.96598102544836283817916841913, 7.21149427744505179683222147621, 7.69931536429066472922525605470, 8.497771804391192154093576901064, 9.310424015638182849224826873304

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