Properties

Label 2-5445-1.1-c1-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.628·2-s − 1.60·4-s − 5-s − 4.14·7-s + 2.26·8-s + 0.628·10-s − 5.40·13-s + 2.60·14-s + 1.78·16-s + 4.14·17-s + 1.25·19-s + 1.60·20-s − 5.21·23-s + 25-s + 3.39·26-s + 6.66·28-s − 7.04·29-s + 4·31-s − 5.65·32-s − 2.60·34-s + 4.14·35-s − 7.21·37-s − 0.788·38-s − 2.26·40-s − 9.55·41-s − 6.66·43-s + 3.27·46-s + ⋯
L(s)  = 1  − 0.444·2-s − 0.802·4-s − 0.447·5-s − 1.56·7-s + 0.800·8-s + 0.198·10-s − 1.49·13-s + 0.696·14-s + 0.447·16-s + 1.00·17-s + 0.288·19-s + 0.359·20-s − 1.08·23-s + 0.200·25-s + 0.665·26-s + 1.25·28-s − 1.30·29-s + 0.718·31-s − 0.999·32-s − 0.446·34-s + 0.701·35-s − 1.18·37-s − 0.127·38-s − 0.358·40-s − 1.49·41-s − 1.01·43-s + 0.482·46-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: \(0\)
Selberg data: \((2,\ 5445,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1630756358\)
\(L(\frac12)\) \(\approx\) \(0.1630756358\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.628T + 2T^{2} \)
7 \( 1 + 4.14T + 7T^{2} \)
13 \( 1 + 5.40T + 13T^{2} \)
17 \( 1 - 4.14T + 17T^{2} \)
19 \( 1 - 1.25T + 19T^{2} \)
23 \( 1 + 5.21T + 23T^{2} \)
29 \( 1 + 7.04T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 7.21T + 37T^{2} \)
41 \( 1 + 9.55T + 41T^{2} \)
43 \( 1 + 6.66T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 5.21T + 59T^{2} \)
61 \( 1 + 8.29T + 61T^{2} \)
67 \( 1 + 13.2T + 67T^{2} \)
71 \( 1 + 5.21T + 71T^{2} \)
73 \( 1 - 2.89T + 73T^{2} \)
79 \( 1 - 1.25T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 19.2T + 97T^{2} \)
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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

−8.146081759667432362001673007445, −7.51034089911066327533799451010, −6.96003042526801767751294495123, −6.01076164879897076116251135997, −5.23886262809815879089510223092, −4.49261288430424508647275262875, −3.53451702802099246623640043197, −3.09224921762201055537611079748, −1.71892685390897720260339807640, −0.22724247734669744897439356825, 0.22724247734669744897439356825, 1.71892685390897720260339807640, 3.09224921762201055537611079748, 3.53451702802099246623640043197, 4.49261288430424508647275262875, 5.23886262809815879089510223092, 6.01076164879897076116251135997, 6.96003042526801767751294495123, 7.51034089911066327533799451010, 8.146081759667432362001673007445

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