Properties

Label 2-1323-1.1-c1-0-15
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 3·5-s − 6·11-s + 4·13-s + 4·16-s + 3·17-s − 2·19-s − 6·20-s + 6·23-s + 4·25-s + 6·29-s + 4·31-s − 7·37-s − 3·41-s − 43-s + 12·44-s + 9·47-s − 8·52-s + 6·53-s − 18·55-s + 9·59-s + 10·61-s − 8·64-s + 12·65-s − 4·67-s − 6·68-s − 2·73-s + ⋯
L(s)  = 1  − 4-s + 1.34·5-s − 1.80·11-s + 1.10·13-s + 16-s + 0.727·17-s − 0.458·19-s − 1.34·20-s + 1.25·23-s + 4/5·25-s + 1.11·29-s + 0.718·31-s − 1.15·37-s − 0.468·41-s − 0.152·43-s + 1.80·44-s + 1.31·47-s − 1.10·52-s + 0.824·53-s − 2.42·55-s + 1.17·59-s + 1.28·61-s − 64-s + 1.48·65-s − 0.488·67-s − 0.727·68-s − 0.234·73-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: $\chi_{1323} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.630538063\)
\(L(\frac12)\) \(\approx\) \(1.630538063\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.757704353671394550269388555688, −8.677781718759589525261607373664, −8.386609491702580500148378137845, −7.19528682144870768370486195193, −6.04248676207121526761624269951, −5.40863841069341263955171468202, −4.79831898195110296666259349467, −3.44521260796395376114982703678, −2.42168483312985269777609133770, −0.980257339797202376885739263498, 0.980257339797202376885739263498, 2.42168483312985269777609133770, 3.44521260796395376114982703678, 4.79831898195110296666259349467, 5.40863841069341263955171468202, 6.04248676207121526761624269951, 7.19528682144870768370486195193, 8.386609491702580500148378137845, 8.677781718759589525261607373664, 9.757704353671394550269388555688

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