Properties

Label 2-1083-3.2-c0-0-1
Degree $2$
Conductor $1083$
Sign $1$
Analytic cond. $0.540487$
Root an. cond. $0.735178$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s − 7-s + 9-s + 12-s − 13-s + 16-s − 21-s + 25-s + 27-s − 28-s − 31-s + 36-s − 37-s − 39-s − 43-s + 48-s − 52-s − 61-s − 63-s + 64-s − 67-s − 73-s + 75-s − 79-s + 81-s − 84-s + ⋯
L(s)  = 1  + 3-s + 4-s − 7-s + 9-s + 12-s − 13-s + 16-s − 21-s + 25-s + 27-s − 28-s − 31-s + 36-s − 37-s − 39-s − 43-s + 48-s − 52-s − 61-s − 63-s + 64-s − 67-s − 73-s + 75-s − 79-s + 81-s − 84-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1083\)    =    \(3 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.540487\)
Root analytic conductor: \(0.735178\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1083} (362, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1083,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.593231853\)
\(L(\frac12)\) \(\approx\) \(1.593231853\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
19 \( 1 \)
good2 \( ( 1 - T )( 1 + T ) \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )^{2} \)
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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

−10.06290955317929830767358687855, −9.288797674257555469372289811718, −8.452967929121358153835405476936, −7.35157812485478151889927165723, −7.02532427698434232535483449218, −6.05528727277577329109096119249, −4.80679393526692214256010234380, −3.45799166917935983993729646004, −2.86640946537598527360617691881, −1.79957236321509578091517513370, 1.79957236321509578091517513370, 2.86640946537598527360617691881, 3.45799166917935983993729646004, 4.80679393526692214256010234380, 6.05528727277577329109096119249, 7.02532427698434232535483449218, 7.35157812485478151889927165723, 8.452967929121358153835405476936, 9.288797674257555469372289811718, 10.06290955317929830767358687855

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