L-function 1-4400-4400.131-r0-0-0

• Label: 1-4400-4400.131-r0-0-0
• Degree: $1$
• Conductor: $4400$
• Sign: $0.990 + 0.140i$
• Analytic cond.: $20.4335$
• Root an. cond.: $20.4335$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.951 − 0.309i)3^-s − 7^-s + (0.809 + 0.587i)9^-s + (−0.587 + 0.809i)13^-s + (−0.309 − 0.951i)17^-s + (−0.951 + 0.309i)19^-s + (0.951 + 0.309i)21^-s + (−0.809 + 0.587i)23^-s + (−0.587 − 0.809i)27^-s + (−0.951 − 0.309i)29^-s + (−0.309 − 0.951i)31^-s + (−0.587 + 0.809i)37^-s + (0.809 − 0.587i)39^-s + (−0.809 − 0.587i)41^-s − i·43^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$4400$    =    $2^{4} \cdot 5^{2} \cdot 11$
Sign:	$0.990 + 0.140i$
Analytic conductor:	$20.4335$
Root analytic conductor:	$20.4335$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4400} (131, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 4400,\ (0:\ ),\ 0.990 + 0.140i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3920915950 + 0.02776157387i$
$L(1)$	$\approx$	$0.5377296194 - 0.03917557612i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	11	$1$
good	3	$1 + (-0.951 - 0.309i)T$
	7	$1 - T$
	13	$1 + (-0.587 + 0.809i)T$
	17	$1 + (-0.309 - 0.951i)T$
	19	$1 + (-0.951 + 0.309i)T$
	23	$1 + (-0.809 + 0.587i)T$
	29	$1 + (-0.951 - 0.309i)T$
	31	$1 + (-0.309 - 0.951i)T$
	37	$1 + (-0.587 + 0.809i)T$

Imaginary part of the first few zeros on the critical line

−18.20012871251016780207506868974, −17.44433041660767593708795773307, −16.996209449865126201004239141560, −16.26732802959076770590628740949, −15.73834439601073372515820088712, −15.02644259405022580941917426914, −14.4428212146136480348924665270, −13.15402746439517321549819014114, −12.81055328308936898087140478890, −12.312009642701794932877201663463, −11.433058170274662396040498029487, −10.566702819539977475974443267508, −10.29690269334647147665407837161, −9.52419778305527882322420085799, −8.73969179725757250233415911906, −7.870996880262008958433249711048, −6.835002458707671021972504234995, −6.49477039327569362323569288747, −5.66800116946514276425609862467, −5.06352045348682826228194963445, −4.093795988423035501938203763, −3.5599013034950681146381107340, −2.53545788719077861008714280553, −1.55335392740191637995372529392, −0.29688652814926654129583513775, 0.3834364995096303362031753394, 1.76478419106179032921927469163, 2.31114154118858867450503631525, 3.52128285214082277138874656898, 4.23888387904412816459517291867, 5.04638160090665722485359287588, 5.83246036159451516909782461303, 6.47179975041017378694540778702, 7.0608376275266414071280063760, 7.68508740819888304442195272384, 8.75306060697034200849272835695, 9.66313622206235889217649970005, 9.96590306534066003599795252723, 10.92156140755795626716985229071, 11.61802186591049585888932800745, 12.12187774956396721286181648827, 12.85628018422539198961256462409, 13.429023180549236368121710015214, 14.10298495331486788982112534796, 15.107188192554149485593132642056, 15.83992587269591521738466035522, 16.32275533466551024417009049512, 17.09066661341142222704726797786, 17.37878849127784845768930471341, 18.498436146798507868473513456945

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