Properties

Label 2-5445-1.1-c1-0-27
Degree $2$
Conductor $5445$
Sign $1$
Analytic cond. $43.4785$
Root an. cond. $6.59382$
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 + 5-s − 7-s + 2·13-s + 4·16-s − 6·17-s − 7·19-s − 2·20-s + 6·23-s + 25-s + 2·28-s − 31-s − 35-s − 7·37-s + 6·41-s + 8·43-s − 6·49-s − 4·52-s + 6·53-s + 12·59-s − 61-s − 8·64-s + 2·65-s − 7·67-s + 12·68-s − 6·71-s − 13·73-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s − 0.377·7-s + 0.554·13-s + 16-s − 1.45·17-s − 1.60·19-s − 0.447·20-s + 1.25·23-s + 1/5·25-s + 0.377·28-s − 0.179·31-s − 0.169·35-s − 1.15·37-s + 0.937·41-s + 1.21·43-s − 6/7·49-s − 0.554·52-s + 0.824·53-s + 1.56·59-s − 0.128·61-s − 64-s + 0.248·65-s − 0.855·67-s + 1.45·68-s − 0.712·71-s − 1.52·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5445\)    =    \(3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(43.4785\)
Root analytic conductor: \(6.59382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5445} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5445,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.200307971\)
\(L(\frac12)\) \(\approx\) \(1.200307971\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−8.485453916851019492858569664820, −7.43711424473686269431816778687, −6.61280187235421899316911179550, −6.05793013905856029986241325619, −5.20779791692855593559951650094, −4.44793309957623836809852653633, −3.87307626331343778310808010397, −2.84635947998777897362121685781, −1.86441514423774169059856966061, −0.58620388806846202709441772114, 0.58620388806846202709441772114, 1.86441514423774169059856966061, 2.84635947998777897362121685781, 3.87307626331343778310808010397, 4.44793309957623836809852653633, 5.20779791692855593559951650094, 6.05793013905856029986241325619, 6.61280187235421899316911179550, 7.43711424473686269431816778687, 8.485453916851019492858569664820

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