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Properties

Label 1-539-539.131-r0-0-0
Degree $1$
Conductor $539$
Sign $0.170 - 0.985i$
Analytic cond. $2.50310$
Root an. cond. $2.50310$
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  + (0.733 − 0.680i)2-s + (0.988 − 0.149i)3-s + (0.0747 − 0.997i)4-s + (−0.365 + 0.930i)5-s + (0.623 − 0.781i)6-s + (−0.623 − 0.781i)8-s + (0.955 − 0.294i)9-s + (0.365 + 0.930i)10-s + (−0.0747 − 0.997i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.900 + 0.433i)20-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)2-s + (0.988 − 0.149i)3-s + (0.0747 − 0.997i)4-s + (−0.365 + 0.930i)5-s + (0.623 − 0.781i)6-s + (−0.623 − 0.781i)8-s + (0.955 − 0.294i)9-s + (0.365 + 0.930i)10-s + (−0.0747 − 0.997i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.900 + 0.433i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.170 - 0.985i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.170 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.066536901 - 1.740311462i\)
\(L(\frac12)\) \(\approx\) \(2.066536901 - 1.740311462i\)
\(L(1)\) \(\approx\) \(1.772047513 - 0.8754007319i\)
\(L(1)\) \(\approx\) \(1.772047513 - 0.8754007319i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.733 - 0.680i)T \)
3 \( 1 + (0.988 - 0.149i)T \)
5 \( 1 + (-0.365 + 0.930i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.826 - 0.563i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.826 + 0.563i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.0747 + 0.997i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (0.0747 - 0.997i)T \)
59 \( 1 + (-0.365 - 0.930i)T \)
61 \( 1 + (0.0747 + 0.997i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (-0.733 - 0.680i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.955 + 0.294i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.64191340767033814989467660134, −23.095183848893473024153219883971, −21.69300449607189881169211630691, −21.14268336470583780611802945633, −20.57708303391690363918230438299, −19.49360671188685129834883984659, −18.81004195272652079836694497730, −17.34033835130596182881897329854, −16.56044273081597831049853933055, −15.92627858494087392609064332302, −15.02430941550383828872197779477, −14.26915257916243762681255950966, −13.57269094060494747211283349806, −12.46243321972737122918011457932, −12.177092639605569589568499781485, −10.59461770125113756624194609192, −9.26003005430839024915521020065, −8.592316740814188062529216512170, −7.85010502470460771667722180772, −6.95194960206113910690966886048, −5.719231662344209098210019351215, −4.53341925852961598620796117663, −4.05098258901938960477126555037, −2.92408227956285186476119664260, −1.63572926588669988646763307026, 1.11911735836766160434628691287, 2.71955874821104353413173225000, 2.92029679978826576238498709691, 4.03101441505939720895755979581, 5.11417459779737216009107077313, 6.461339498698959869198829524160, 7.28913252188460856456041091296, 8.30370531885727084484192866042, 9.59823911255603025931498214190, 10.236682911337535336603158897529, 11.21219759292040100569734524867, 12.12458671307744747105440288090, 13.13646611283904673766994841807, 13.742900708659261110043516910477, 14.845736880846566567397312441982, 15.03424147560424274151334183552, 16.00566447490530285266231435573, 17.726240934406620405616565001, 18.56149481107058557911579261815, 19.2967534220582608435328493838, 19.84565036930028625320293646112, 20.734613298341608007160206908719, 21.52162731544861024145731947267, 22.31367261415106910386934733965, 23.197169897644324872020957477129

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