Properties

Label 2-539-77.4-c1-0-27
Degree $2$
Conductor $539$
Sign $-0.934 + 0.355i$
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  + (0.153 − 1.45i)2-s + (−1.08 − 1.20i)3-s + (−0.147 − 0.0314i)4-s + (0.426 + 0.189i)5-s + (−1.91 + 1.39i)6-s + (0.837 − 2.57i)8-s + (0.0399 − 0.379i)9-s + (0.342 − 0.592i)10-s + (3.24 − 0.702i)11-s + (0.122 + 0.212i)12-s + (1.28 + 0.930i)13-s + (−0.233 − 0.718i)15-s + (−3.90 − 1.74i)16-s + (−0.546 − 5.19i)17-s + (−0.547 − 0.116i)18-s + (−4.12 + 0.877i)19-s + ⋯
L(s)  = 1  + (0.108 − 1.03i)2-s + (−0.625 − 0.694i)3-s + (−0.0739 − 0.0157i)4-s + (0.190 + 0.0848i)5-s + (−0.783 + 0.569i)6-s + (0.296 − 0.911i)8-s + (0.0133 − 0.126i)9-s + (0.108 − 0.187i)10-s + (0.977 − 0.211i)11-s + (0.0353 + 0.0612i)12-s + (0.355 + 0.257i)13-s + (−0.0602 − 0.185i)15-s + (−0.977 − 0.435i)16-s + (−0.132 − 1.26i)17-s + (−0.129 − 0.0274i)18-s + (−0.947 + 0.201i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.934 + 0.355i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (312, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.934 + 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.254268 - 1.38192i\)
\(L(\frac12)\) \(\approx\) \(0.254268 - 1.38192i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.24 + 0.702i)T \)
good2 \( 1 + (-0.153 + 1.45i)T + (-1.95 - 0.415i)T^{2} \)
3 \( 1 + (1.08 + 1.20i)T + (-0.313 + 2.98i)T^{2} \)
5 \( 1 + (-0.426 - 0.189i)T + (3.34 + 3.71i)T^{2} \)
13 \( 1 + (-1.28 - 0.930i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.546 + 5.19i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (4.12 - 0.877i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (-0.902 - 1.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.840 - 2.58i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.18 + 0.526i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (1.29 - 1.44i)T + (-3.86 - 36.7i)T^{2} \)
41 \( 1 + (0.321 - 0.990i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 + (-6.25 + 1.32i)T + (42.9 - 19.1i)T^{2} \)
53 \( 1 + (12.0 - 5.37i)T + (35.4 - 39.3i)T^{2} \)
59 \( 1 + (-8.41 - 1.78i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + (13.9 + 6.19i)T + (40.8 + 45.3i)T^{2} \)
67 \( 1 + (-2.33 + 4.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.88 - 5.72i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-13.0 - 2.76i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (0.374 - 3.56i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (-13.9 + 10.1i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-4.45 - 7.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.18 - 1.58i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.80469229286745780224035390726, −9.684215989356814433359148246085, −8.981719632673129108144757334333, −7.50795268526911176336448548202, −6.63506549400016755070870166415, −6.05429260474016117696394485012, −4.47698635949292506549324584293, −3.40304552982139192796015876065, −2.06434745704399047337030681013, −0.885175997855087187211923853604, 1.96993248920826240887472295463, 3.95848006847904764582771738215, 4.82019966566250431176651926113, 5.92179244437198583331563841398, 6.33558927616438048911792576988, 7.51456376714614511083516306148, 8.430231528741335010844795209112, 9.352022906921390308620806351455, 10.54834320261716290073991120468, 10.95674650049700541753572125357

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