L-function 1-287-287.251-r1-0-0

• Label: 1-287-287.251-r1-0-0
• Degree: $1$
• Conductor: $287$
• Sign: $-0.389 + 0.920i$
• Analytic cond.: $30.8424$
• Root an. cond.: $30.8424$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.309 + 0.951i)2^-s − i·3^-s + (−0.809 − 0.587i)4^-s + (−0.809 − 0.587i)5^-s + (0.951 + 0.309i)6^-s + (0.809 − 0.587i)8^-s − 9^-s + (0.809 − 0.587i)10^-s + (−0.587 − 0.809i)11^-s + (−0.587 + 0.809i)12^-s + (−0.951 − 0.309i)13^-s + (−0.587 + 0.809i)15^-s + (0.309 + 0.951i)16^-s + (−0.587 − 0.809i)17^-s + (0.309 − 0.951i)18^-s + (−0.951 + 0.309i)19^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$287$    =    $7 \cdot 41$
Sign:	$-0.389 + 0.920i$
Analytic conductor:	$30.8424$
Root analytic conductor:	$30.8424$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{287} (251, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 287,\ (1:\ ),\ -0.389 + 0.920i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.02066773276 + 0.03119814572i$
$L(1)$	$\approx$	$0.5282765150 - 0.1096462889i$

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	7	$1$
	41	$1$
good	2	$1 + (-0.309 + 0.951i)T$
	3	$1 - iT$
	5	$1 + (-0.809 - 0.587i)T$
	11	$1 + (-0.587 - 0.809i)T$
	13	$1 + (-0.951 - 0.309i)T$
	17	$1 + (-0.587 - 0.809i)T$
	19	$1 + (-0.951 + 0.309i)T$
	23	$1 + (0.309 - 0.951i)T$
	29	$1 + (0.587 - 0.809i)T$

Imaginary part of the first few zeros on the critical line

−26.2618288371138569108278464005, −25.54961804520272489362543645191, −23.64165565409548231714646403766, −23.03664031661121009105192034401, −21.96588791567275962474291493270, −21.55307649921809511523385480853, −20.369536081133056384568720761753, −19.686539712772585211877664663098, −18.9661489495003411027809453310, −17.64763003688306205866918181235, −17.02900288614142827186272113869, −15.64202368798136138747286051610, −15.01048149879606891316574612592, −13.92991114076731747577077184860, −12.53663915937768998193972532309, −11.745184089856135070260853138504, −10.66539958151823095189533793057, −10.25564999934999370812723826972, −9.06188210486932779510738496456, −8.138241777277962834325095149313, −6.939774611981438685836135388985, −5.00322201293056704211648334289, −4.27010668845106669949050576465, −3.22211247068561373552604307447, −2.19426289228481967050272913474, 0.018550613062762027050536024969, 0.76587859352790840308647731221, 2.59473790514954902161892399996, 4.37198551606160517797497939082, 5.40752614971410083779084095659, 6.52355112591131929073065131536, 7.51445403985258711971261054104, 8.24166921052364505074934305975, 8.960661453641624206648728229813, 10.48082081193990480392659763850, 11.730035793131516981402409593881, 12.76747130206111533800589152523, 13.51751543130544333997876402834, 14.591972237485250752851005908647, 15.549751663850667546423227134186, 16.53408295070467286861380541658, 17.25670464153243512483762778530, 18.27276853854396189115327217184, 19.16211719540853663266890891209, 19.6562925651200838007481976496, 20.92645121702869329314724931416, 22.55962473224938814245343922434, 23.1084981196675410394782849823, 24.20209087445321641251444855063, 24.43370746381753036295774193246

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