Properties

Label 2-5445-1.1-c1-0-143
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  + 0.485·2-s − 1.76·4-s + 5-s + 1.61·7-s − 1.82·8-s + 0.485·10-s − 4.95·13-s + 0.785·14-s + 2.64·16-s − 3.13·17-s − 1.09·19-s − 1.76·20-s + 6.75·23-s + 25-s − 2.40·26-s − 2.85·28-s + 3.55·29-s + 8.53·31-s + 4.93·32-s − 1.52·34-s + 1.61·35-s − 5.35·37-s − 0.532·38-s − 1.82·40-s − 5.32·41-s − 3.41·43-s + 3.27·46-s + ⋯
L(s)  = 1  + 0.343·2-s − 0.882·4-s + 0.447·5-s + 0.612·7-s − 0.645·8-s + 0.153·10-s − 1.37·13-s + 0.210·14-s + 0.660·16-s − 0.761·17-s − 0.251·19-s − 0.394·20-s + 1.40·23-s + 0.200·25-s − 0.471·26-s − 0.540·28-s + 0.659·29-s + 1.53·31-s + 0.872·32-s − 0.261·34-s + 0.273·35-s − 0.881·37-s − 0.0864·38-s − 0.288·40-s − 0.830·41-s − 0.520·43-s + 0.483·46-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 \( 1 - 0.485T + 2T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
13 \( 1 + 4.95T + 13T^{2} \)
17 \( 1 + 3.13T + 17T^{2} \)
19 \( 1 + 1.09T + 19T^{2} \)
23 \( 1 - 6.75T + 23T^{2} \)
29 \( 1 - 3.55T + 29T^{2} \)
31 \( 1 - 8.53T + 31T^{2} \)
37 \( 1 + 5.35T + 37T^{2} \)
41 \( 1 + 5.32T + 41T^{2} \)
43 \( 1 + 3.41T + 43T^{2} \)
47 \( 1 + 6.60T + 47T^{2} \)
53 \( 1 + 9.71T + 53T^{2} \)
59 \( 1 + 7.18T + 59T^{2} \)
61 \( 1 - 6.63T + 61T^{2} \)
67 \( 1 - 2.36T + 67T^{2} \)
71 \( 1 - 16.6T + 71T^{2} \)
73 \( 1 + 8.49T + 73T^{2} \)
79 \( 1 - 1.99T + 79T^{2} \)
83 \( 1 - 9.54T + 83T^{2} \)
89 \( 1 - 2.65T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−8.095772431763871685957720716500, −6.88368876907183993910884570733, −6.47594927286509350994039494781, −5.23515229403851740266501308510, −4.95971796231689262912655263427, −4.39398464156087907481280880529, −3.24443655032619481375607007964, −2.49117928694064696206761568527, −1.34291201171161684247745558790, 0, 1.34291201171161684247745558790, 2.49117928694064696206761568527, 3.24443655032619481375607007964, 4.39398464156087907481280880529, 4.95971796231689262912655263427, 5.23515229403851740266501308510, 6.47594927286509350994039494781, 6.88368876907183993910884570733, 8.095772431763871685957720716500

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