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Properties

Label 2-539-7.2-c1-0-1
Degree $2$
Conductor $539$
Sign $-0.991 - 0.126i$
Analytic cond. $4.30393$
Root an. cond. $2.07459$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.917 + 1.58i)2-s + (−1.09 − 1.90i)3-s + (−0.682 − 1.18i)4-s + (−0.317 + 0.550i)5-s + 4.03·6-s − 1.16·8-s + (−0.917 + 1.58i)9-s + (−0.582 − 1.00i)10-s + (0.5 + 0.866i)11-s + (−1.50 + 2.59i)12-s + 1.80·13-s + 1.39·15-s + (2.43 − 4.21i)16-s + (−1.41 − 2.45i)17-s + (−1.68 − 2.91i)18-s + (−2.78 + 4.81i)19-s + ⋯
L(s)  = 1  + (−0.648 + 1.12i)2-s + (−0.634 − 1.09i)3-s + (−0.341 − 0.590i)4-s + (−0.142 + 0.246i)5-s + 1.64·6-s − 0.412·8-s + (−0.305 + 0.529i)9-s + (−0.184 − 0.319i)10-s + (0.150 + 0.261i)11-s + (−0.433 + 0.750i)12-s + 0.499·13-s + 0.360·15-s + (0.608 − 1.05i)16-s + (−0.343 − 0.595i)17-s + (−0.396 − 0.686i)18-s + (−0.638 + 1.10i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0188464 + 0.296980i\)
\(L(\frac12)\) \(\approx\) \(0.0188464 + 0.296980i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.917 - 1.58i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.09 + 1.90i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.317 - 0.550i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 1.80T + 13T^{2} \)
17 \( 1 + (1.41 + 2.45i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.78 - 4.81i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.08 - 1.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 + (-3.21 - 5.56i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.03 - 5.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.53T + 41T^{2} \)
43 \( 1 + 4.86T + 43T^{2} \)
47 \( 1 + (1.41 - 2.45i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.73 + 6.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.90 - 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.16 - 3.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.801 - 1.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.29T + 71T^{2} \)
73 \( 1 + (7.99 + 13.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.38 - 4.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.23T + 83T^{2} \)
89 \( 1 + (-0.182 + 0.315i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.59T + 97T^{2} \)
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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

−11.43288744734166322459439063081, −10.23756816918726713226408585364, −9.181394458828236872588876580140, −8.285636417147341035136391514136, −7.44546202855511561199342645296, −6.83297538037669983814970283823, −6.16472156396585058150240399188, −5.24943084960689706720339839075, −3.46857356793797381248522087489, −1.59340196262523469650623882310, 0.22956608322202638051537193765, 2.05734830037307223644655200373, 3.55884120221955186042830647746, 4.41125488555231929367480925272, 5.58465398367939372955789009345, 6.56867011139984345106985625630, 8.229296702892827166179617337428, 8.988596190999637796723744286250, 9.731302239777891896356561659652, 10.55570355300192529106039792231

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