Properties

Label 2-539-1.1-c1-0-22
Degree $2$
Conductor $539$
Sign $1$
Analytic cond. $4.30393$
Root an. cond. $2.07459$
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  + 1.87·2-s + 0.652·3-s + 1.53·4-s + 3.53·5-s + 1.22·6-s − 0.879·8-s − 2.57·9-s + 6.63·10-s + 11-s + 12-s + 4.41·13-s + 2.30·15-s − 4.71·16-s − 5.24·17-s − 4.83·18-s + 1.81·19-s + 5.41·20-s + 1.87·22-s − 6.33·23-s − 0.573·24-s + 7.47·25-s + 8.29·26-s − 3.63·27-s + 1.92·29-s + 4.33·30-s + 1.46·31-s − 7.10·32-s + ⋯
L(s)  = 1  + 1.32·2-s + 0.376·3-s + 0.766·4-s + 1.57·5-s + 0.500·6-s − 0.310·8-s − 0.857·9-s + 2.09·10-s + 0.301·11-s + 0.288·12-s + 1.22·13-s + 0.595·15-s − 1.17·16-s − 1.27·17-s − 1.14·18-s + 0.416·19-s + 1.21·20-s + 0.400·22-s − 1.32·23-s − 0.117·24-s + 1.49·25-s + 1.62·26-s − 0.700·27-s + 0.356·29-s + 0.791·30-s + 0.263·31-s − 1.25·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.558364933\)
\(L(\frac12)\) \(\approx\) \(3.558364933\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 - T \)
good2 \( 1 - 1.87T + 2T^{2} \)
3 \( 1 - 0.652T + 3T^{2} \)
5 \( 1 - 3.53T + 5T^{2} \)
13 \( 1 - 4.41T + 13T^{2} \)
17 \( 1 + 5.24T + 17T^{2} \)
19 \( 1 - 1.81T + 19T^{2} \)
23 \( 1 + 6.33T + 23T^{2} \)
29 \( 1 - 1.92T + 29T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 - 4.45T + 37T^{2} \)
41 \( 1 - 0.283T + 41T^{2} \)
43 \( 1 + 3.41T + 43T^{2} \)
47 \( 1 + 4.55T + 47T^{2} \)
53 \( 1 + 7.23T + 53T^{2} \)
59 \( 1 - 9.53T + 59T^{2} \)
61 \( 1 + 1.14T + 61T^{2} \)
67 \( 1 + 0.694T + 67T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 - 2.34T + 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−11.07610577684918782863648438307, −9.884353127172674007247948200321, −9.070234680289049851660992534269, −8.345481287677819273620689181043, −6.53755656665418312317099221670, −6.09708071507662225365662524352, −5.31262599921936768386931761341, −4.13196420143855307039189995532, −2.98306147523787437364138532829, −1.98196403479871147831307618055, 1.98196403479871147831307618055, 2.98306147523787437364138532829, 4.13196420143855307039189995532, 5.31262599921936768386931761341, 6.09708071507662225365662524352, 6.53755656665418312317099221670, 8.345481287677819273620689181043, 9.070234680289049851660992534269, 9.884353127172674007247948200321, 11.07610577684918782863648438307

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