Properties

Label 2-5445-1.1-c1-0-112
Degree $2$
Conductor $5445$
Sign $-1$
Analytic cond. $43.4785$
Root an. cond. $6.59382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.75·2-s + 5.57·4-s + 5-s − 2.73·7-s − 9.83·8-s − 2.75·10-s + 2.77·13-s + 7.51·14-s + 15.9·16-s + 3.78·17-s + 4.25·19-s + 5.57·20-s − 1.37·23-s + 25-s − 7.63·26-s − 15.2·28-s − 8.86·29-s + 5.46·31-s − 24.1·32-s − 10.4·34-s − 2.73·35-s − 2.30·37-s − 11.7·38-s − 9.83·40-s − 8.61·41-s − 12.3·43-s + 3.78·46-s + ⋯
L(s)  = 1  − 1.94·2-s + 2.78·4-s + 0.447·5-s − 1.03·7-s − 3.47·8-s − 0.870·10-s + 0.769·13-s + 2.00·14-s + 3.98·16-s + 0.917·17-s + 0.975·19-s + 1.24·20-s − 0.287·23-s + 0.200·25-s − 1.49·26-s − 2.87·28-s − 1.64·29-s + 0.981·31-s − 4.27·32-s − 1.78·34-s − 0.461·35-s − 0.378·37-s − 1.89·38-s − 1.55·40-s − 1.34·41-s − 1.89·43-s + 0.558·46-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: no
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 \( 1 + 2.75T + 2T^{2} \)
7 \( 1 + 2.73T + 7T^{2} \)
13 \( 1 - 2.77T + 13T^{2} \)
17 \( 1 - 3.78T + 17T^{2} \)
19 \( 1 - 4.25T + 19T^{2} \)
23 \( 1 + 1.37T + 23T^{2} \)
29 \( 1 + 8.86T + 29T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 + 2.30T + 37T^{2} \)
41 \( 1 + 8.61T + 41T^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
47 \( 1 + 5.59T + 47T^{2} \)
53 \( 1 + 0.543T + 53T^{2} \)
59 \( 1 - 0.389T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 - 3.65T + 67T^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 + 8.16T + 73T^{2} \)
79 \( 1 - 3.44T + 79T^{2} \)
83 \( 1 - 4.65T + 83T^{2} \)
89 \( 1 - 6.62T + 89T^{2} \)
97 \( 1 + 16.5T + 97T^{2} \)
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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

−8.106333532293413909323140175930, −7.13017375468220447468138933902, −6.71324022639154903095618920097, −5.97007560439171089169752846180, −5.34967246815699937845995411833, −3.49393407967379481907829488582, −3.11796742844885354663612765858, −1.94506499917752255694545523059, −1.17231364599469402066425997924, 0, 1.17231364599469402066425997924, 1.94506499917752255694545523059, 3.11796742844885354663612765858, 3.49393407967379481907829488582, 5.34967246815699937845995411833, 5.97007560439171089169752846180, 6.71324022639154903095618920097, 7.13017375468220447468138933902, 8.106333532293413909323140175930

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