L-function 1-6384-6384.2189-r0-0-0

• Label: 1-6384-6384.2189-r0-0-0
• Degree: $1$
• Conductor: $6384$
• Sign: $-0.121 - 0.992i$
• Analytic cond.: $29.6471$
• Root an. cond.: $29.6471$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.342 + 0.939i)5^-s + (0.866 − 0.5i)11^-s + (−0.342 + 0.939i)13^-s + (−0.939 + 0.342i)17^-s + (0.766 − 0.642i)23^-s + (−0.766 + 0.642i)25^-s + (−0.642 − 0.766i)29^-s − 31^-s + (−0.866 + 0.5i)37^-s + (0.939 − 0.342i)41^-s + (−0.984 − 0.173i)43^-s + (−0.939 − 0.342i)47^-s + (0.342 − 0.939i)53^-s + (0.766 + 0.642i)55^-s + (0.342 + 0.939i)59^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$6384$    =    $2^{4} \cdot 3 \cdot 7 \cdot 19$
Sign:	$-0.121 - 0.992i$
Analytic conductor:	$29.6471$
Root analytic conductor:	$29.6471$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6384} (2189, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 6384,\ (0:\ ),\ -0.121 - 0.992i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.4662453905 - 0.5269505514i$
$L(1)$	$\approx$	$0.9558000470 + 0.1010594018i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	7	$1$
	19	$1$
good	5	$1 + (0.342 + 0.939i)T$
	11	$1 + (0.866 - 0.5i)T$
	13	$1 + (-0.342 + 0.939i)T$
	17	$1 + (-0.939 + 0.342i)T$
	23	$1 + (0.766 - 0.642i)T$
	29	$1 + (-0.642 - 0.766i)T$
	31	$1 - T$
	37	$1 + (-0.866 + 0.5i)T$
	41	$1 + (0.939 - 0.342i)T$

Imaginary part of the first few zeros on the critical line

−17.6792125397163929868178505502, −17.213577574432438552730440817334, −16.501160698668030013539113642926, −15.93752443421367666925901795084, −15.08717900213415791188645550398, −14.665501264716076401634079782425, −13.73933548251814349614263485527, −13.12576736147537297824219338940, −12.62812053692602003407159083912, −12.01624937452434466169429615438, −11.20798042759701620475206553153, −10.57941216245719750532226977964, −9.579949962100734165632105681714, −9.2705549883071176573015923409, −8.64024759897286640938293441556, −7.77506571975872074172787201566, −7.102988326752761375667350682348, −6.3845940516251856823497389486, −5.42293646112012869106927058052, −5.07280246701298380161701312462, −4.23120523247503168744688502813, −3.50757198917976411184727787172, −2.53914710045531469768877390285, −1.68453021429681951272052578120, −1.05326427289405714783201247683, 0.16887524851565332066632896918, 1.632465371928830476991736584972, 2.076082042366001182785007565221, 3.03151848973861442338713399467, 3.72919758819183521919493536616, 4.41476480138543973439279357133, 5.32265705832249479485101169585, 6.209292392158776273816515977019, 6.68205349598990967791624516336, 7.166598743413610503571115017293, 8.11986844827280446674930969027, 9.035859402556777742764350500652, 9.32817239690129494709067522936, 10.27057139028639891565547598541, 10.89854578842619726166106764904, 11.45678344918447697695457388047, 12.01426434450932611165628012728, 13.014752927715572328977190738962, 13.60408924948158458276631307858, 14.19537256436219097625424792240, 14.89224795243751071016516180012, 15.18418715879445664445064017603, 16.30107112649832884303245925243, 16.791130879481297691802728001502, 17.40854655089489444927031352644

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