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Properties

Label 2-6552-1.1-c1-0-54
Degree $2$
Conductor $6552$
Sign $-1$
Analytic cond. $52.3179$
Root an. cond. $7.23311$
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  − 4·5-s + 7-s + 4·11-s + 13-s − 6·17-s + 4·19-s − 6·23-s + 11·25-s − 4·35-s − 6·37-s − 12·41-s + 4·43-s + 6·47-s + 49-s + 8·53-s − 16·55-s + 14·59-s − 14·61-s − 4·65-s + 4·67-s + 12·71-s + 6·73-s + 4·77-s + 8·79-s + 6·83-s + 24·85-s − 16·89-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.377·7-s + 1.20·11-s + 0.277·13-s − 1.45·17-s + 0.917·19-s − 1.25·23-s + 11/5·25-s − 0.676·35-s − 0.986·37-s − 1.87·41-s + 0.609·43-s + 0.875·47-s + 1/7·49-s + 1.09·53-s − 2.15·55-s + 1.82·59-s − 1.79·61-s − 0.496·65-s + 0.488·67-s + 1.42·71-s + 0.702·73-s + 0.455·77-s + 0.900·79-s + 0.658·83-s + 2.60·85-s − 1.69·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6552\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(52.3179\)
Root analytic conductor: \(7.23311\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6552,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
13 \( 1 - T \)
good5 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−7.67946373898693430872713009623, −6.95861657329754046805309677656, −6.54128786423930256583665635842, −5.39121180457450983911503457908, −4.57997435167887026446300777382, −3.86397384587159733531445839396, −3.59530623488060000364580708400, −2.30170108376473064534505446417, −1.14628760189264047011801830476, 0, 1.14628760189264047011801830476, 2.30170108376473064534505446417, 3.59530623488060000364580708400, 3.86397384587159733531445839396, 4.57997435167887026446300777382, 5.39121180457450983911503457908, 6.54128786423930256583665635842, 6.95861657329754046805309677656, 7.67946373898693430872713009623

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