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Properties

Label 1-936-936.133-r0-0-0
Degree $1$
Conductor $936$
Sign $-0.815 - 0.578i$
Analytic cond. $4.34676$
Root an. cond. $4.34676$
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)5-s + (−0.5 + 0.866i)7-s − 11-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)35-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s − 11-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)35-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.815 - 0.578i$
Analytic conductor: \(4.34676\)
Root analytic conductor: \(4.34676\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 936,\ (0:\ ),\ -0.815 - 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1690722138 - 0.5302945334i\)
\(L(\frac12)\) \(\approx\) \(0.1690722138 - 0.5302945334i\)
\(L(1)\) \(\approx\) \(0.8141407557 - 0.1594676672i\)
\(L(1)\) \(\approx\) \(0.8141407557 - 0.1594676672i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 - T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−22.130721943914030633772867723181, −21.51265316235213133838451515555, −20.52615640549353995911817915839, −19.80336679328615476846651757665, −18.97441237154922801113804589487, −18.165375479016938055009117685588, −17.49811647779052736598630582019, −16.7064762371497842614406200632, −15.69728648227310840322335811719, −15.05659882744662493199290807881, −14.07617930597489789862174179462, −13.27533836871873034254651474652, −12.92237404518463458045322669558, −11.31870394456924924278648070071, −10.918320026106288366986399938910, −9.92426203404675750270116943263, −9.46451145051558923891370899138, −7.988238679579262591459767737781, −7.35795783047434483269117248946, −6.44376720448813890834162958389, −5.70248495235182515196632049998, −4.51479849629492212889655735224, −3.450410562896791954117256845950, −2.67542852104119795442956715746, −1.52964582032484596115303425539, 0.22338692048638912535715542669, 1.83481665266733743495427606098, 2.59097103634675358710750019285, 3.7826544010600686552470855150, 5.09159402210146862718316083017, 5.494096351793957270532135163024, 6.47524992939188506412974146596, 7.644251141102224456390465623179, 8.55617966774511799258530575848, 9.27295572049419221469413510122, 9.98435143905003370333366064284, 10.97025447720194028430250764938, 12.175327022521884765877033865181, 12.58093295241475049823837680053, 13.43613913993651008452191539639, 14.219027953420790386654122514867, 15.33741426326920265464586899139, 16.123234157315502397243965283249, 16.50382919019695761867017335454, 17.73319701080079905142035117096, 18.33150066551152562332020575276, 19.01247534198063656485230962450, 20.27197552829516353278748885071, 20.57724160808289595759410531116, 21.53818398907808500373312643466

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