L-function 1-800-800.629-r0-0-0

• Label: 1-800-800.629-r0-0-0
• Degree: $1$
• Conductor: $800$
• Sign: $-0.331 - 0.943i$
• Analytic cond.: $3.71518$
• Root an. cond.: $3.71518$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.891 − 0.453i)3^-s − i·7^-s + (0.587 + 0.809i)9^-s + (−0.156 − 0.987i)11^-s + (−0.156 + 0.987i)13^-s + (0.309 − 0.951i)17^-s + (0.453 + 0.891i)19^-s + (−0.453 + 0.891i)21^-s + (0.587 − 0.809i)23^-s + (−0.156 − 0.987i)27^-s + (0.891 + 0.453i)29^-s + (0.309 − 0.951i)31^-s + (−0.309 + 0.951i)33^-s + (−0.987 − 0.156i)37^-s + (0.587 − 0.809i)39^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$800$    =    $2^{5} \cdot 5^{2}$
Sign:	$-0.331 - 0.943i$
Analytic conductor:	$3.71518$
Root analytic conductor:	$3.71518$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{800} (629, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 800,\ (0:\ ),\ -0.331 - 0.943i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.5279318094 - 0.7449354096i$
$L(1)$	$\approx$	$0.7539374220 - 0.2927869470i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
good	3	$1 + (-0.891 - 0.453i)T$
	7	$1 - iT$
	11	$1 + (-0.156 - 0.987i)T$
	13	$1 + (-0.156 + 0.987i)T$
	17	$1 + (0.309 - 0.951i)T$
	19	$1 + (0.453 + 0.891i)T$
	23	$1 + (0.587 - 0.809i)T$
	29	$1 + (0.891 + 0.453i)T$
	31	$1 + (0.309 - 0.951i)T$

Imaginary part of the first few zeros on the critical line

−22.612531122665488343362348400855, −21.60448864778609257904451196299, −21.27475776071164834969761768815, −20.143916487543657554396839078354, −19.30756175251815256294377393992, −18.21861506951178502463793631117, −17.63770409281876040909238357828, −17.0986797060466236666072535697, −15.74891034959308498814999406947, −15.46677979412118070189147674352, −14.75575644045736597016059830279, −13.34266329326651104184909899846, −12.38752897563980931613339050428, −12.04089276669868275273035076653, −10.9307739846948891621898351189, −10.17756669251636438345763642291, −9.41451366278607075855043417324, −8.43337575869009700802013547891, −7.28498369335555412023319808746, −6.36902971844700910864884460611, −5.3423582875734425829334604991, −4.95267755020258788123865547920, −3.6534216664630974253575831599, −2.57089877929035843637937447098, −1.2193647102278179015678039881, 0.54773580249756213204996447375, 1.524110900524368834703853684490, 2.946837142117452838484802801991, 4.17003970807921295089267089222, 5.01239188959979762771775252863, 6.02625695187220645284402392633, 6.869535911940970050690070250202, 7.54478807044398288240494021908, 8.57171160547698379189708420160, 9.83328001696619630321382337347, 10.57006177213013339699075810434, 11.41426870806100928327485580876, 12.01605961325462581079750376572, 13.08855880258771015875902935652, 13.82904598407940585018760864264, 14.42265263961545236316626575359, 16.02704436689718849018928088703, 16.44345425494558599843890712655, 17.0559463385575655323361043547, 18.02830107514704893858629905186, 18.82005360636978312773057638253, 19.34979124558373517139559031990, 20.56606127747037414585296962670, 21.212732184154016016637029267659, 22.208076267817735491470388658168

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