L-function 1-664-664.587-r0-0-0

• Label: 1-664-664.587-r0-0-0
• Degree: $1$
• Conductor: $664$
• Sign: $0.701 + 0.713i$
• Analytic cond.: $3.08360$
• Root an. cond.: $3.08360$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.817 + 0.575i)3^-s + (−0.973 − 0.227i)5^-s + (−0.720 − 0.693i)7^-s + (0.338 + 0.941i)9^-s + (−0.665 + 0.746i)11^-s + (0.953 + 0.301i)13^-s + (−0.665 − 0.746i)15^-s + (0.606 − 0.795i)17^-s + (0.543 − 0.839i)19^-s + (−0.190 − 0.981i)21^-s + (0.859 − 0.511i)23^-s + (0.896 + 0.443i)25^-s + (−0.264 + 0.964i)27^-s + (0.409 + 0.912i)29^-s + (−0.477 + 0.878i)31^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$664$    =    $2^{3} \cdot 83$
Sign:	$0.701 + 0.713i$
Analytic conductor:	$3.08360$
Root analytic conductor:	$3.08360$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{664} (587, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 664,\ (0:\ ),\ 0.701 + 0.713i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.342746876 + 0.5628013016i$
$L(1)$	$\approx$	$1.132505642 + 0.2171023585i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	83	$1$
good	3	$1 + (0.817 + 0.575i)T$
	5	$1 + (-0.973 - 0.227i)T$
	7	$1 + (-0.720 - 0.693i)T$
	11	$1 + (-0.665 + 0.746i)T$
	13	$1 + (0.953 + 0.301i)T$
	17	$1 + (0.606 - 0.795i)T$
	19	$1 + (0.543 - 0.839i)T$
	23	$1 + (0.859 - 0.511i)T$
	29	$1 + (0.409 + 0.912i)T$

Imaginary part of the first few zeros on the critical line

−23.033388064834274304508339304206, −21.79230860976569131703288932253, −20.92995871009456466487743962804, −20.16441897666196387675525853932, −19.10714899324906352956060710067, −18.88893676184646632665985550475, −18.22379479478792657036323567189, −16.81218480680909125215183926120, −15.70255158598990499243021105794, −15.44175556009411760054496105327, −14.4066444875237693255786898294, −13.43554466899645256279685688944, −12.72317072158842486922270437421, −11.9667829519644686479807470283, −10.98326044282360331920370789860, −9.903361143427290331946044662707, −8.799036972205050480624763703177, −8.17328818561573638260030238069, −7.46618018259297158436062163321, −6.3278443345139137532404473243, −5.535740360059333413972463190, −3.66579321744007151170266617867, −3.43386976066447729319182761746, −2.30016222986481637561213688316, −0.8076899765169809588799925825, 1.09776601807613288762541739461, 2.833208217284892684282414981994, 3.43787759883869431959566656558, 4.43568163582495583407343537322, 5.15905267207824472007179062562, 6.987905197460742269989704580695, 7.382646253489651779471132383477, 8.546064719693625906753956905807, 9.193651947695783415169873551063, 10.26518453871446541976334030922, 10.88708068117042629350150515693, 12.06374120821353562408879593208, 13.061248658400515139339736741649, 13.70725753918080189953149816697, 14.71437081650746580015264731298, 15.63669658331393560816090296203, 16.07523781994191328219262054476, 16.78207256144683757483945863288, 18.22059863561722345765352664605, 18.98323440476213017336790542839, 19.83932312941541837537275073769, 20.401675368082220470462790816470, 20.94828421206284922344624807846, 22.130021343059734014109115187910, 23.04311855361144210729078119919

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