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Properties

Label 2-1539-171.94-c0-0-1
Degree $2$
Conductor $1539$
Sign $0.342 - 0.939i$
Analytic cond. $0.768061$
Root an. cond. $0.876390$
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  + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)7-s + (0.866 + 1.5i)11-s + (−0.499 − 0.866i)16-s + 1.73·17-s − 19-s + (−0.866 + 1.5i)23-s + (0.5 + 0.866i)25-s + 0.999·28-s + (0.5 + 0.866i)43-s − 1.73·44-s + (0.866 + 1.5i)47-s + (−0.5 − 0.866i)61-s + 0.999·64-s + (−0.866 + 1.49i)68-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)7-s + (0.866 + 1.5i)11-s + (−0.499 − 0.866i)16-s + 1.73·17-s − 19-s + (−0.866 + 1.5i)23-s + (0.5 + 0.866i)25-s + 0.999·28-s + (0.5 + 0.866i)43-s − 1.73·44-s + (0.866 + 1.5i)47-s + (−0.5 − 0.866i)61-s + 0.999·64-s + (−0.866 + 1.49i)68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1539\)    =    \(3^{4} \cdot 19\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(0.768061\)
Root analytic conductor: \(0.876390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1539} (1405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1539,\ (\ :0),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9245582693\)
\(L(\frac12)\) \(\approx\) \(0.9245582693\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - 1.73T + T^{2} \)
23 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.615049896102339303361393768204, −9.255443786835798736848025211102, −7.991133270728189839965774216850, −7.46585859149892347307458194621, −6.84195063223407042454993404256, −5.69334749812455195810878536391, −4.50488726368350385894000464028, −3.92042578708394677997915128936, −3.10067121046940258699200218270, −1.51764448042115562281317412595, 0.824422451095514847737657661584, 2.34846176695093962251067272920, 3.52840476023300113603433004857, 4.46034451524519144630286133610, 5.78695024001869949576116599347, 5.89115900950352381093697150404, 6.80120056357639509219199999653, 8.334776428000769903287920940308, 8.655894728680968975348738428578, 9.422199216561214184337520120617

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