L-function 1-2664-2664.43-r0-0-0

• Label: 1-2664-2664.43-r0-0-0
• Degree: $1$
• Conductor: $2664$
• Sign: $0.00469 + 0.999i$
• Analytic cond.: $12.3715$
• Root an. cond.: $12.3715$
• 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 + (0.5 + 0.866i)7^-s + (0.5 + 0.866i)11^-s + (0.866 + 0.5i)13^-s + i·17^-s + i·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 + i·35^-s + (0.5 − 0.866i)41^-s + (0.866 − 0.5i)43^-s + (0.5 + 0.866i)47^-s + (−0.5 + 0.866i)49^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00469 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2664$    =    $2^{3} \cdot 3^{2} \cdot 37$
Sign:	$0.00469 + 0.999i$
Analytic conductor:	$12.3715$
Root analytic conductor:	$12.3715$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2664} (43, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2664,\ (0:\ ),\ 0.00469 + 0.999i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.766382431 + 1.774690022i$
$L(1)$	$\approx$	$1.372776208 + 0.5322521132i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−19.172310619432261037198595602573, −18.197233197653730697823916005632, −17.748426500740988504756933390039, −17.06239514496007555173808014830, −16.32932021507523304153970076821, −15.880411696455929182792362564648, −14.59559940450336301352088835776, −14.152488723605171009756884880219, −13.35589358882140892664366748127, −13.05755237470214472412182970208, −11.930062950959864106024411289144, −11.0165027269347288081844288130, −10.7028359639166754338951224287, −9.68536924495506663695115101613, −8.88363697988471527216513570843, −8.472733989029010354345460494069, −7.35415452305924157999303563802, −6.657323030963331929297736748220, −5.83322282399164938105185183726, −5.02304656785010401996367021096, −4.39351936620470391090841114634, −3.30601925785161356518528587980, −2.54590682213914660467419765622, −1.17000475798230121632656774398, −0.914178274836057461204262265511, 1.54064663794710485615933296089, 1.79405877476479143741865021213, 2.82569627987802659236938358951, 3.82098070699647417158316387455, 4.64944611480502380781021099129, 5.78574694874915221056812607296, 6.02344593542957554977914662875, 6.97969968244603073749773779563, 7.83176256242074759617698476368, 8.80410191462067532725137665438, 9.290583925213968427040470564312, 10.112842020238915007694498969924, 10.899070249666062524045004990761, 11.52605930375484759270836658879, 12.4831027862986491254630370843, 12.939032961079911495822919620971, 14.13669041516455420072330819573, 14.347313092829538888458399636124, 15.222219735173119536510809079871, 15.7793971369093025734200102035, 16.93145422455185461884695097167, 17.434469084061645949081382882343, 18.013032921878938258335767715579, 18.89840443692165663860271517793, 19.12036079400943297994765235167

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