Properties

Label 2-5445-1.1-c1-0-137
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  − 1.22·2-s − 0.496·4-s + 5-s + 0.451·7-s + 3.06·8-s − 1.22·10-s + 4.84·13-s − 0.553·14-s − 2.75·16-s + 0.740·17-s − 6.80·19-s − 0.496·20-s + 0.00634·23-s + 25-s − 5.93·26-s − 0.224·28-s − 0.323·29-s − 5.60·31-s − 2.73·32-s − 0.907·34-s + 0.451·35-s + 7.36·37-s + 8.34·38-s + 3.06·40-s − 10.9·41-s + 1.80·43-s − 0.00777·46-s + ⋯
L(s)  = 1  − 0.866·2-s − 0.248·4-s + 0.447·5-s + 0.170·7-s + 1.08·8-s − 0.387·10-s + 1.34·13-s − 0.148·14-s − 0.689·16-s + 0.179·17-s − 1.56·19-s − 0.111·20-s + 0.00132·23-s + 0.200·25-s − 1.16·26-s − 0.0424·28-s − 0.0601·29-s − 1.00·31-s − 0.484·32-s − 0.155·34-s + 0.0763·35-s + 1.21·37-s + 1.35·38-s + 0.484·40-s − 1.71·41-s + 0.275·43-s − 0.00114·46-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 \( 1 + 1.22T + 2T^{2} \)
7 \( 1 - 0.451T + 7T^{2} \)
13 \( 1 - 4.84T + 13T^{2} \)
17 \( 1 - 0.740T + 17T^{2} \)
19 \( 1 + 6.80T + 19T^{2} \)
23 \( 1 - 0.00634T + 23T^{2} \)
29 \( 1 + 0.323T + 29T^{2} \)
31 \( 1 + 5.60T + 31T^{2} \)
37 \( 1 - 7.36T + 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 - 1.80T + 43T^{2} \)
47 \( 1 - 1.67T + 47T^{2} \)
53 \( 1 + 8.63T + 53T^{2} \)
59 \( 1 - 1.50T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 - 9.60T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 - 2.09T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 - 9.36T + 89T^{2} \)
97 \( 1 - 5.87T + 97T^{2} \)
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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

−8.087901176938627706740627975770, −7.25316349674001385244538219938, −6.38855748394985896599545809815, −5.81656650742432476828017085355, −4.81407121659196876472828596826, −4.16587165280233467532222428297, −3.26096373148554855695672178063, −1.96373093320871666731923724602, −1.31119115966279728506136636598, 0, 1.31119115966279728506136636598, 1.96373093320871666731923724602, 3.26096373148554855695672178063, 4.16587165280233467532222428297, 4.81407121659196876472828596826, 5.81656650742432476828017085355, 6.38855748394985896599545809815, 7.25316349674001385244538219938, 8.087901176938627706740627975770

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