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Properties

Label 2-539-77.37-c1-0-10
Degree $2$
Conductor $539$
Sign $0.610 - 0.792i$
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  + (0.206 − 0.0920i)2-s + (−0.214 + 0.0455i)3-s + (−1.30 + 1.44i)4-s + (−0.260 − 2.48i)5-s + (−0.0401 + 0.0291i)6-s + (−0.276 + 0.849i)8-s + (−2.69 + 1.20i)9-s + (−0.282 − 0.488i)10-s + (3.04 + 1.32i)11-s + (0.213 − 0.369i)12-s + (4.15 + 3.01i)13-s + (0.168 + 0.520i)15-s + (−0.386 − 3.67i)16-s + (1.31 + 0.584i)17-s + (−0.446 + 0.496i)18-s + (4.06 + 4.50i)19-s + ⋯
L(s)  = 1  + (0.146 − 0.0650i)2-s + (−0.123 + 0.0263i)3-s + (−0.652 + 0.724i)4-s + (−0.116 − 1.10i)5-s + (−0.0163 + 0.0118i)6-s + (−0.0975 + 0.300i)8-s + (−0.898 + 0.400i)9-s + (−0.0892 − 0.154i)10-s + (0.917 + 0.398i)11-s + (0.0616 − 0.106i)12-s + (1.15 + 0.837i)13-s + (0.0436 + 0.134i)15-s + (−0.0965 − 0.918i)16-s + (0.318 + 0.141i)17-s + (−0.105 + 0.116i)18-s + (0.931 + 1.03i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.610 - 0.792i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.610 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07052 + 0.526646i\)
\(L(\frac12)\) \(\approx\) \(1.07052 + 0.526646i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.04 - 1.32i)T \)
good2 \( 1 + (-0.206 + 0.0920i)T + (1.33 - 1.48i)T^{2} \)
3 \( 1 + (0.214 - 0.0455i)T + (2.74 - 1.22i)T^{2} \)
5 \( 1 + (0.260 + 2.48i)T + (-4.89 + 1.03i)T^{2} \)
13 \( 1 + (-4.15 - 3.01i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.31 - 0.584i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (-4.06 - 4.50i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (3.54 - 6.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.01 - 6.19i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.803 + 7.64i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-3.89 - 0.828i)T + (33.8 + 15.0i)T^{2} \)
41 \( 1 + (2.08 - 6.41i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.802T + 43T^{2} \)
47 \( 1 + (4.51 + 5.01i)T + (-4.91 + 46.7i)T^{2} \)
53 \( 1 + (0.687 - 6.54i)T + (-51.8 - 11.0i)T^{2} \)
59 \( 1 + (1.92 - 2.13i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (0.0894 + 0.850i)T + (-59.6 + 12.6i)T^{2} \)
67 \( 1 + (-0.823 - 1.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.65 - 2.65i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-9.93 + 11.0i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (-2.24 + 0.997i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (-1.81 + 1.32i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.867 + 1.50i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.77 + 7.09i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.34986487385351470076622060383, −9.763030775311627276691268161711, −9.085859647170210180478966994207, −8.366881834505209077774287097133, −7.67488360883258157856469741962, −6.16398964943557090545292734748, −5.22302754812373909554285455703, −4.23473867819641094969618223753, −3.40944567351422751937113188908, −1.43644342192566348988739840122, 0.78820129092985081810972016912, 2.92634622937277461900253519142, 3.82587719963899001740291718061, 5.22564533474304318998239663168, 6.21240208364377643723839845669, 6.65627400527421240313654454665, 8.183360226862100681785353226681, 8.929561816005725412340799944227, 9.891437493093581047971610674337, 10.78754048237426826551177968897

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