Properties

Label 2-5445-1.1-c1-0-52
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  + 0.414·2-s − 1.82·4-s + 5-s + 4.82·7-s − 1.58·8-s + 0.414·10-s − 5.65·13-s + 1.99·14-s + 3·16-s − 6.82·17-s + 1.17·19-s − 1.82·20-s + 4·23-s + 25-s − 2.34·26-s − 8.82·28-s + 0.828·29-s + 4.41·32-s − 2.82·34-s + 4.82·35-s + 0.343·37-s + 0.485·38-s − 1.58·40-s − 0.828·41-s + 3.17·43-s + 1.65·46-s + 4·47-s + ⋯
L(s)  = 1  + 0.292·2-s − 0.914·4-s + 0.447·5-s + 1.82·7-s − 0.560·8-s + 0.130·10-s − 1.56·13-s + 0.534·14-s + 0.750·16-s − 1.65·17-s + 0.268·19-s − 0.408·20-s + 0.834·23-s + 0.200·25-s − 0.459·26-s − 1.66·28-s + 0.153·29-s + 0.780·32-s − 0.485·34-s + 0.816·35-s + 0.0564·37-s + 0.0787·38-s − 0.250·40-s − 0.129·41-s + 0.483·43-s + 0.244·46-s + 0.583·47-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

\(L(1)\) \(\approx\) \(2.123680983\)
\(L(\frac12)\) \(\approx\) \(2.123680983\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.414T + 2T^{2} \)
7 \( 1 - 4.82T + 7T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 + 6.82T + 17T^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 0.343T + 37T^{2} \)
41 \( 1 + 0.828T + 41T^{2} \)
43 \( 1 - 3.17T + 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 0.343T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 + 10T + 83T^{2} \)
89 \( 1 - 7.65T + 89T^{2} \)
97 \( 1 - 0.343T + 97T^{2} \)
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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

−8.247678890309799175778647832503, −7.48925896783488924545672001575, −6.84122328744642054211108932895, −5.68263231865689236678490589789, −5.06886606576379787740390285001, −4.66881151650890024491299705208, −4.03369758628296482486033243557, −2.65799673594129466341023519489, −1.99360234043674250341966951984, −0.75955528901466713275009357434, 0.75955528901466713275009357434, 1.99360234043674250341966951984, 2.65799673594129466341023519489, 4.03369758628296482486033243557, 4.66881151650890024491299705208, 5.06886606576379787740390285001, 5.68263231865689236678490589789, 6.84122328744642054211108932895, 7.48925896783488924545672001575, 8.247678890309799175778647832503

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