Properties

Label 2-5544-1.1-c1-0-68
Degree $2$
Conductor $5544$
Sign $-1$
Analytic cond. $44.2690$
Root an. cond. $6.65350$
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  + 5-s + 7-s − 11-s + 3·13-s + 2·17-s − 5·19-s − 8·23-s − 4·25-s + 7·29-s − 8·31-s + 35-s − 3·37-s − 10·41-s − 10·43-s + 7·47-s + 49-s − 2·53-s − 55-s − 9·59-s − 2·61-s + 3·65-s − 3·67-s + 6·71-s + 73-s − 77-s + 10·79-s − 6·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.301·11-s + 0.832·13-s + 0.485·17-s − 1.14·19-s − 1.66·23-s − 4/5·25-s + 1.29·29-s − 1.43·31-s + 0.169·35-s − 0.493·37-s − 1.56·41-s − 1.52·43-s + 1.02·47-s + 1/7·49-s − 0.274·53-s − 0.134·55-s − 1.17·59-s − 0.256·61-s + 0.372·65-s − 0.366·67-s + 0.712·71-s + 0.117·73-s − 0.113·77-s + 1.12·79-s − 0.658·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5544\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(44.2690\)
Root analytic conductor: \(6.65350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5544} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5544,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 + T \)
good5 \( 1 - T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.073084630431476484476689233571, −7.01527169594557496639414462714, −6.26987639071685409062447047312, −5.72425661295894586935645977308, −4.93145239966570525178861675874, −4.04826239329640791793521723312, −3.34560287012376144813849282539, −2.14797581069306532413213929434, −1.55576962732955691364002151558, 0, 1.55576962732955691364002151558, 2.14797581069306532413213929434, 3.34560287012376144813849282539, 4.04826239329640791793521723312, 4.93145239966570525178861675874, 5.72425661295894586935645977308, 6.26987639071685409062447047312, 7.01527169594557496639414462714, 8.073084630431476484476689233571

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