Properties

Label 2-5445-1.1-c1-0-132
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 5-s + 2·7-s + 3·8-s − 10-s − 4·13-s − 2·14-s − 16-s − 6·17-s + 6·19-s − 20-s − 4·23-s + 25-s + 4·26-s − 2·28-s + 6·29-s + 8·31-s − 5·32-s + 6·34-s + 2·35-s − 6·37-s − 6·38-s + 3·40-s − 6·41-s + 6·43-s + 4·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.755·7-s + 1.06·8-s − 0.316·10-s − 1.10·13-s − 0.534·14-s − 1/4·16-s − 1.45·17-s + 1.37·19-s − 0.223·20-s − 0.834·23-s + 1/5·25-s + 0.784·26-s − 0.377·28-s + 1.11·29-s + 1.43·31-s − 0.883·32-s + 1.02·34-s + 0.338·35-s − 0.986·37-s − 0.973·38-s + 0.474·40-s − 0.937·41-s + 0.914·43-s + 0.589·46-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: \(1\)
Selberg data: \((2,\ 5445,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.017793086743598694281416112863, −7.24125778469533790692442723144, −6.57006303177130665405311043825, −5.51841037900744680559944025854, −4.71826128070397951740716941416, −4.48379582514255780675005499843, −3.11551783391647075558634882317, −2.11305576069239070091241887537, −1.27112542139718822332489563723, 0, 1.27112542139718822332489563723, 2.11305576069239070091241887537, 3.11551783391647075558634882317, 4.48379582514255780675005499843, 4.71826128070397951740716941416, 5.51841037900744680559944025854, 6.57006303177130665405311043825, 7.24125778469533790692442723144, 8.017793086743598694281416112863

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