Properties

Label 2-5445-1.1-c1-0-86
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  − 1.35·2-s − 0.162·4-s + 5-s − 3.64·7-s + 2.93·8-s − 1.35·10-s − 2.83·13-s + 4.94·14-s − 3.64·16-s − 3.69·17-s − 0.0951·19-s − 0.162·20-s − 1.16·23-s + 25-s + 3.83·26-s + 0.591·28-s + 6.75·29-s + 6.77·31-s − 0.914·32-s + 5.00·34-s − 3.64·35-s + 9.83·37-s + 0.128·38-s + 2.93·40-s + 8.31·41-s − 2.96·43-s + 1.57·46-s + ⋯
L(s)  = 1  − 0.958·2-s − 0.0810·4-s + 0.447·5-s − 1.37·7-s + 1.03·8-s − 0.428·10-s − 0.785·13-s + 1.32·14-s − 0.912·16-s − 0.895·17-s − 0.0218·19-s − 0.0362·20-s − 0.242·23-s + 0.200·25-s + 0.752·26-s + 0.111·28-s + 1.25·29-s + 1.21·31-s − 0.161·32-s + 0.858·34-s − 0.616·35-s + 1.61·37-s + 0.0209·38-s + 0.463·40-s + 1.29·41-s − 0.452·43-s + 0.232·46-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: Trivial
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 \( 1 + 1.35T + 2T^{2} \)
7 \( 1 + 3.64T + 7T^{2} \)
13 \( 1 + 2.83T + 13T^{2} \)
17 \( 1 + 3.69T + 17T^{2} \)
19 \( 1 + 0.0951T + 19T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 - 6.75T + 29T^{2} \)
31 \( 1 - 6.77T + 31T^{2} \)
37 \( 1 - 9.83T + 37T^{2} \)
41 \( 1 - 8.31T + 41T^{2} \)
43 \( 1 + 2.96T + 43T^{2} \)
47 \( 1 - 2.22T + 47T^{2} \)
53 \( 1 + 2.99T + 53T^{2} \)
59 \( 1 - 8.50T + 59T^{2} \)
61 \( 1 + 8.48T + 61T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 + 8.30T + 71T^{2} \)
73 \( 1 + 1.32T + 73T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + 4.33T + 97T^{2} \)
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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

−7.899603583885823142487238473422, −7.18570614478777731686175811206, −6.46270764671842175988882361140, −5.92697210091665191598725503928, −4.72301207883392371487380939029, −4.26071990054352459930152276522, −2.98258343218293954963616812830, −2.34285631463948532861335358908, −1.02595641029530689625875798968, 0, 1.02595641029530689625875798968, 2.34285631463948532861335358908, 2.98258343218293954963616812830, 4.26071990054352459930152276522, 4.72301207883392371487380939029, 5.92697210091665191598725503928, 6.46270764671842175988882361140, 7.18570614478777731686175811206, 7.899603583885823142487238473422

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