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Properties

Label 2-539-77.59-c0-0-0
Degree $2$
Conductor $539$
Sign $0.442 - 0.896i$
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  + (−0.0646 + 0.614i)2-s + (0.604 + 0.128i)4-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.604 − 0.128i)18-s + 0.618·22-s + (−0.309 − 0.535i)23-s + (0.669 + 0.743i)25-s + (−0.5 − 1.53i)29-s + (−0.5 + 0.866i)32-s + (−0.190 + 0.587i)36-s + (0.413 − 0.459i)37-s − 1.61·43-s + (0.0646 − 0.614i)44-s + ⋯
L(s)  = 1  + (−0.0646 + 0.614i)2-s + (0.604 + 0.128i)4-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.604 − 0.128i)18-s + 0.618·22-s + (−0.309 − 0.535i)23-s + (0.669 + 0.743i)25-s + (−0.5 − 1.53i)29-s + (−0.5 + 0.866i)32-s + (−0.190 + 0.587i)36-s + (0.413 − 0.459i)37-s − 1.61·43-s + (0.0646 − 0.614i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.442 - 0.896i$
Analytic conductor: \(0.268996\)
Root analytic conductor: \(0.518648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :0),\ 0.442 - 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9887662984\)
\(L(\frac12)\) \(\approx\) \(0.9887662984\)
\(L(1)\) 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 \( 1 + (0.104 + 0.994i)T \)
good2 \( 1 + (0.0646 - 0.614i)T + (-0.978 - 0.207i)T^{2} \)
3 \( 1 + (0.104 - 0.994i)T^{2} \)
5 \( 1 + (-0.669 - 0.743i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.978 - 0.207i)T^{2} \)
19 \( 1 + (-0.913 + 0.406i)T^{2} \)
23 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.669 + 0.743i)T^{2} \)
37 \( 1 + (-0.413 + 0.459i)T + (-0.104 - 0.994i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + (-0.913 + 0.406i)T^{2} \)
53 \( 1 + (1.47 - 0.658i)T + (0.669 - 0.743i)T^{2} \)
59 \( 1 + (-0.913 - 0.406i)T^{2} \)
61 \( 1 + (-0.669 - 0.743i)T^{2} \)
67 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.913 - 0.406i)T^{2} \)
79 \( 1 + (-0.169 + 1.60i)T + (-0.978 - 0.207i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−11.12192627435723269445406781936, −10.54002517622214046576982147515, −9.257687116781282834406191299096, −8.165539001274853788119371260955, −7.77913644712386319723562038111, −6.61054435883460989108076894674, −5.82056401381821857722274880484, −4.87121119046073565200332816518, −3.28545146148892397233808562247, −2.11378030800102174452325582218, 1.52887838484128004326729690525, 2.86839505731870621128647741785, 3.89644361129510181897526199837, 5.25870108013756722964982456292, 6.54245242742443332365021345376, 7.02799609225419603483035291557, 8.294919788878767742121251059992, 9.480732228017032978566268582482, 9.970192230733074100040662977948, 10.94843731810528004439406920777

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