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Properties

Label 2-531-531.58-c0-0-1
Degree $2$
Conductor $531$
Sign $0.173 - 0.984i$
Analytic cond. $0.265003$
Root an. cond. $0.514784$
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  + (0.766 + 0.642i)3-s + (−0.5 + 0.866i)4-s + (−0.173 + 0.300i)5-s + (−0.173 − 0.300i)7-s + (0.173 + 0.984i)9-s + (−0.939 + 0.342i)12-s + (−0.326 + 0.118i)15-s + (−0.499 − 0.866i)16-s − 17-s + 1.53·19-s + (−0.173 − 0.300i)20-s + (0.0603 − 0.342i)21-s + (0.439 + 0.761i)25-s + (−0.500 + 0.866i)27-s + 0.347·28-s + (−0.766 − 1.32i)29-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)3-s + (−0.5 + 0.866i)4-s + (−0.173 + 0.300i)5-s + (−0.173 − 0.300i)7-s + (0.173 + 0.984i)9-s + (−0.939 + 0.342i)12-s + (−0.326 + 0.118i)15-s + (−0.499 − 0.866i)16-s − 17-s + 1.53·19-s + (−0.173 − 0.300i)20-s + (0.0603 − 0.342i)21-s + (0.439 + 0.761i)25-s + (−0.500 + 0.866i)27-s + 0.347·28-s + (−0.766 − 1.32i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(0.265003\)
Root analytic conductor: \(0.514784\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :0),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9625757844\)
\(L(\frac12)\) \(\approx\) \(0.9625757844\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.766 - 0.642i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 - 1.53T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - 1.53T + T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
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

−11.21570036766810636854030815256, −10.23169198080426868921101056406, −9.294696291198538452372450920694, −8.777718748979147175080797255366, −7.63648285642755782356583537984, −7.18857271836033045875945847175, −5.45276097069587951354496097017, −4.28680775453627923024663788888, −3.58027024032222140686178573570, −2.54332082411237830170823845884, 1.29625866088587057971035083904, 2.76072340715358078544189768871, 4.14345465947699023055714842689, 5.27991997753689012688690785910, 6.32498534164911005241967262511, 7.26884746138833973785441270530, 8.362038887795026844367470500271, 9.123627751127032645370841562697, 9.657641099677085625649231294689, 10.81651819912537340958540710825

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