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Properties

Label 2-539-77.32-c0-0-0
Degree $2$
Conductor $539$
Sign $-0.749 - 0.661i$
Analytic cond. $0.268996$
Root an. cond. $0.518648$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 1.22i)3-s + (−0.5 + 0.866i)4-s + (0.707 + 1.22i)5-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.707 − 1.22i)12-s − 2·15-s + (−0.499 − 0.866i)16-s − 1.41·20-s + (−0.499 + 0.866i)25-s + (0.707 − 1.22i)31-s + (0.707 + 1.22i)33-s + 0.999·36-s + (1 + 1.73i)37-s + (0.499 + 0.866i)44-s + (0.707 − 1.22i)45-s + ⋯
L(s)  = 1  + (−0.707 + 1.22i)3-s + (−0.5 + 0.866i)4-s + (0.707 + 1.22i)5-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.707 − 1.22i)12-s − 2·15-s + (−0.499 − 0.866i)16-s − 1.41·20-s + (−0.499 + 0.866i)25-s + (0.707 − 1.22i)31-s + (0.707 + 1.22i)33-s + 0.999·36-s + (1 + 1.73i)37-s + (0.499 + 0.866i)44-s + (0.707 − 1.22i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.749 - 0.661i$
Analytic conductor: \(0.268996\)
Root analytic conductor: \(0.518648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :0),\ -0.749 - 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7146915602\)
\(L(\frac12)\) \(\approx\) \(0.7146915602\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−11.39302270512994538929157931891, −10.38070668861945467022415277965, −9.822946125573690109713482062603, −8.964646276107471146431718153014, −7.86838544084029834651876577212, −6.61065756825603917344511360816, −5.85474293071912250319986279292, −4.69198850770971458884436522489, −3.72081401633552309534950811031, −2.80007744783019375201042596444, 1.07376047675134623727043302673, 1.91011127188030000481789348213, 4.42976042574493672296418647129, 5.25417347155450699073075554197, 6.04854341977163214808321033495, 6.81518468095392877351405953092, 8.000617016425726869859252538013, 9.126837966366779678333294023235, 9.623156831817113718438606825910, 10.71990092100064096810566379044

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