L-function 1-612-612.259-r1-0-0

• Label: 1-612-612.259-r1-0-0
• Degree: $1$
• Conductor: $612$
• Sign: $-0.883 - 0.469i$
• Analytic cond.: $65.7685$
• Root an. cond.: $65.7685$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.866 + 0.5i)5^-s + (−0.866 + 0.5i)7^-s + (−0.866 + 0.5i)11^-s + (−0.5 + 0.866i)13^-s + 19^-s + (0.866 + 0.5i)23^-s + (0.5 + 0.866i)25^-s + (−0.866 + 0.5i)29^-s + (−0.866 − 0.5i)31^-s − 35^-s − i·37^-s + (−0.866 − 0.5i)41^-s + (−0.5 − 0.866i)43^-s + (0.5 + 0.866i)47^-s + (0.5 − 0.866i)49^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.883 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$612$    =    $2^{2} \cdot 3^{2} \cdot 17$
Sign:	$-0.883 - 0.469i$
Analytic conductor:	$65.7685$
Root analytic conductor:	$65.7685$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{612} (259, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 612,\ (1:\ ),\ -0.883 - 0.469i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-0.09540903430 + 0.3829227012i$
$L(1)$	$\approx$	$0.8642033524 + 0.2486131522i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	17	$1$
good	5	$1 + (0.866 + 0.5i)T$
	7	$1 + (-0.866 + 0.5i)T$
	11	$1 + (-0.866 + 0.5i)T$
	13	$1 + (-0.5 + 0.866i)T$
	19	$1 + T$
	23	$1 + (0.866 + 0.5i)T$
	29	$1 + (-0.866 + 0.5i)T$
	31	$1 + (-0.866 - 0.5i)T$
	37	$1 - iT$

Imaginary part of the first few zeros on the critical line

−22.27722225008360645258683555530, −21.63021481773371080194382256049, −20.46740690904437191251664841560, −20.21718313688049738329146104510, −18.96663154430347965171437668336, −18.226595665923756420735575787730, −17.25457737152801886140641507301, −16.585270894133360241028650610828, −15.8400900865394222626826683677, −14.79300396229439119284800818296, −13.66534821927820368448035720636, −13.13575108920864980301880800235, −12.50210778095989812373848261177, −11.16030316689218070228092404835, −10.160549009182770409788380167498, −9.6879310338161606527267113782, −8.61109078303353629791404001071, −7.597392565034497180064213514474, −6.5857262917475046974087130556, −5.55595375583413445180849417050, −4.92608250293959050017162146290, −3.40162806959528072872605841623, −2.62954488011918370167237005876, −1.170805268832044619818929205325, −0.09346994510256607597638448387, 1.74785483355286004305356905976, 2.615378352226111321764645484822, 3.55374823670906440066982328067, 5.10431585826206078664828861690, 5.73068881491345424481963492641, 6.88297043560690482251916436541, 7.454552534922015809391116455233, 9.09986469493126946795685808900, 9.51208772997280266348464180285, 10.40791477533569089930881052059, 11.38140132109734101936181381347, 12.498798078492825558637046209197, 13.17973210964391064811974530881, 14.044143204499813208667675647615, 14.949908264912354986277897778543, 15.77207083416223112898620457668, 16.69301683282679888678570886260, 17.5389331389498053093660420586, 18.54281185515901890473426822992, 18.872124305454373339684334369422, 20.06818302492790155698797181235, 20.93155662010394644613006003158, 21.82657516027108233285983256504, 22.30536174491850706658187718268, 23.20378185194730403310954907503

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