L-function 1-3895-3895.1104-r0-0-0

• Label: 1-3895-3895.1104-r0-0-0
• Degree: $1$
• Conductor: $3895$
• Sign: $-0.127 + 0.991i$
• Analytic cond.: $18.0883$
• Root an. cond.: $18.0883$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.342 + 0.939i)2^-s + (−0.573 + 0.819i)3^-s + (−0.766 − 0.642i)4^-s + (−0.573 − 0.819i)6^-s + (0.965 − 0.258i)7^-s + (0.866 − 0.5i)8^-s + (−0.342 − 0.939i)9^-s + (−0.258 + 0.965i)11^-s + (0.965 − 0.258i)12^-s + (0.819 − 0.573i)13^-s + (−0.0871 + 0.996i)14^-s + (0.173 + 0.984i)16^-s + (0.906 − 0.422i)17^-s + 18^-s + (−0.342 + 0.939i)21^-s + (−0.819 − 0.573i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3895$    =    $5 \cdot 19 \cdot 41$
Sign:	$-0.127 + 0.991i$
Analytic conductor:	$18.0883$
Root analytic conductor:	$18.0883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3895} (1104, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3895,\ (0:\ ),\ -0.127 + 0.991i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.9446443880 + 1.073502777i$
$L(1)$	$\approx$	$0.7309744864 + 0.5195160700i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	19	$1$
	41	$1$
good	2	$1 + (-0.342 + 0.939i)T$
	3	$1 + (-0.573 + 0.819i)T$
	7	$1 + (0.965 - 0.258i)T$
	11	$1 + (-0.258 + 0.965i)T$
	13	$1 + (0.819 - 0.573i)T$
	17	$1 + (0.906 - 0.422i)T$
	23	$1 + (0.766 + 0.642i)T$
	29	$1 + (0.906 + 0.422i)T$
	31	$1 + (-0.5 + 0.866i)T$

Imaginary part of the first few zeros on the critical line

−18.56776850776520816408381207072, −17.861691498876344370218064894587, −17.14258515863112232872987990177, −16.65062125499055599412088903271, −15.90011415735848500738546437794, −14.587101082657434307601536646129, −14.073470855034313711252493810017, −13.334340508214676818733635086054, −12.8035614057581927325542933801, −11.8606686720111336372035225915, −11.59405931615650263680532046640, −10.78169326756230483791977737691, −10.4832206282855192995020888196, −9.17270300526112144968913414396, −8.56502728462775306522908923691, −7.96442373946593924731102120796, −7.37923000111453738598575013917, −6.168429529689694972748046220761, −5.630470280950399879895914656972, −4.728898181495080637207488720154, −3.97743912037292396497311996887, −2.87918436927242420142864665650, −2.225856317279043212539513227291, −1.25621894962851756373535923051, −0.81749971160555952941474199401, 0.808141853015280256810744204334, 1.49987051701935414155327158138, 3.028425439773584372852057391525, 4.00614545966310265309167155389, 4.68797652436323483728688593109, 5.31187962576355995218222788257, 5.77353150711191843837651663435, 6.86714615327209640042474431838, 7.40892807507863876830123215426, 8.281558210874579375568921393130, 8.86989865327686136014751970438, 9.78939027835855986294149556457, 10.25192061293447946263266436098, 10.94969704535485239236131465847, 11.634207439811977649328459902329, 12.605721440861108537147514367920, 13.36241091480453206164497460118, 14.43128967196463685252811088569, 14.65151782328397198776980326319, 15.45764349572796045983406318788, 16.003927323620580283415984735736, 16.6292869330555580138911881655, 17.37677142009691207131979379858, 17.97514738664905977047941784940, 18.13907423974083039324331645841

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