Properties

Label 2-5445-1.1-c1-0-105
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  − 2.33·2-s + 3.44·4-s − 5-s − 1.04·7-s − 3.38·8-s + 2.33·10-s + 5.71·13-s + 2.44·14-s + 1.00·16-s + 1.04·17-s + 4.66·19-s − 3.44·20-s − 4.89·23-s + 25-s − 13.3·26-s − 3.61·28-s − 2.57·29-s − 2·31-s + 4.43·32-s − 2.44·34-s + 1.04·35-s − 6.89·37-s − 10.8·38-s + 3.38·40-s − 6.76·41-s + 1.04·43-s + 11.4·46-s + ⋯
L(s)  = 1  − 1.65·2-s + 1.72·4-s − 0.447·5-s − 0.396·7-s − 1.19·8-s + 0.738·10-s + 1.58·13-s + 0.654·14-s + 0.250·16-s + 0.254·17-s + 1.07·19-s − 0.771·20-s − 1.02·23-s + 0.200·25-s − 2.61·26-s − 0.684·28-s − 0.477·29-s − 0.359·31-s + 0.783·32-s − 0.420·34-s + 0.177·35-s − 1.13·37-s − 1.76·38-s + 0.535·40-s − 1.05·41-s + 0.160·43-s + 1.68·46-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 \( 1 + 2.33T + 2T^{2} \)
7 \( 1 + 1.04T + 7T^{2} \)
13 \( 1 - 5.71T + 13T^{2} \)
17 \( 1 - 1.04T + 17T^{2} \)
19 \( 1 - 4.66T + 19T^{2} \)
23 \( 1 + 4.89T + 23T^{2} \)
29 \( 1 + 2.57T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 6.89T + 37T^{2} \)
41 \( 1 + 6.76T + 41T^{2} \)
43 \( 1 - 1.04T + 43T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 7.23T + 61T^{2} \)
67 \( 1 - 8.89T + 67T^{2} \)
71 \( 1 + 1.10T + 71T^{2} \)
73 \( 1 - 1.52T + 73T^{2} \)
79 \( 1 - 9.80T + 79T^{2} \)
83 \( 1 - 3.61T + 83T^{2} \)
89 \( 1 + 3.79T + 89T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
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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

−7.82357760864887982601025970522, −7.50264015760261618646016929355, −6.52523844280314460439435556741, −6.05918369383212241082345996136, −4.99045895173595680234393165349, −3.75649584370811358765198395695, −3.21726047307933092892602328440, −1.92345488943174495927667102536, −1.12733855695365154973184879866, 0, 1.12733855695365154973184879866, 1.92345488943174495927667102536, 3.21726047307933092892602328440, 3.75649584370811358765198395695, 4.99045895173595680234393165349, 6.05918369383212241082345996136, 6.52523844280314460439435556741, 7.50264015760261618646016929355, 7.82357760864887982601025970522

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