L-function 1-161-161.4-r0-0-0

• Label: 1-161-161.4-r0-0-0
• Degree: $1$
• Conductor: $161$
• Sign: $0.947 - 0.320i$
• Analytic cond.: $0.747680$
• Root an. cond.: $0.747680$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.327 − 0.945i)2^-s + (−0.888 − 0.458i)3^-s + (−0.786 + 0.618i)4^-s + (−0.995 − 0.0950i)5^-s + (−0.142 + 0.989i)6^-s + (0.841 + 0.540i)8^-s + (0.580 + 0.814i)9^-s + (0.235 + 0.971i)10^-s + (−0.327 + 0.945i)11^-s + (0.981 − 0.189i)12^-s + (−0.959 − 0.281i)13^-s + (0.841 + 0.540i)15^-s + (0.235 − 0.971i)16^-s + (0.928 − 0.371i)17^-s + (0.580 − 0.814i)18^-s + (0.928 + 0.371i)19^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$161$    =    $7 \cdot 23$
Sign:	$0.947 - 0.320i$
Analytic conductor:	$0.747680$
Root analytic conductor:	$0.747680$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{161} (4, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 161,\ (0:\ ),\ 0.947 - 0.320i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.4444831322 - 0.07319836231i$
$L(1)$	$\approx$	$0.4967947139 - 0.1921229643i$

Euler product

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

	$p$	$F_p(T)$
bad	7	$1$
	23	$1$
good	2	$1 + (-0.327 - 0.945i)T$
	3	$1 + (-0.888 - 0.458i)T$
	5	$1 + (-0.995 - 0.0950i)T$
	11	$1 + (-0.327 + 0.945i)T$
	13	$1 + (-0.959 - 0.281i)T$
	17	$1 + (0.928 - 0.371i)T$
	19	$1 + (0.928 + 0.371i)T$
	29	$1 + (-0.142 + 0.989i)T$
	31	$1 + (0.0475 - 0.998i)T$

Imaginary part of the first few zeros on the critical line

−27.63710881360641665026030290144, −26.77524949613548987758512674497, −26.36109355873267439799663802905, −24.64183623106670727507882055743, −23.94459736831697007433785713767, −23.15178135993098855201508453842, −22.32353627872556585660932854201, −21.2923176298778364032690359626, −19.62803569899240331965058563471, −18.803744486563328137461081402, −17.75684196328991547021852372852, −16.65545733385219860905948354478, −16.08178023740117306037701770501, −15.18637905233816162378147777200, −14.15857054175797566898839184069, −12.58221926258976903933862358748, −11.47517670762567687325409445057, −10.41691345891664362134696635107, −9.310879967515698871336084487992, −7.9660724061808513790139088894, −7.041723003446637288857639330135, −5.76637047016295462509932961899, −4.815040324187256546266633097557, −3.59443369598516828505014631397, −0.6053853140338135203592392313, 1.109935004023501036979564997170, 2.772024575296615499754179228056, 4.34445074952192864022662650591, 5.28987913647980880926743852165, 7.35950554948446423204960233685, 7.81315177669100327625292076589, 9.58278185863060822690038150367, 10.536691018880181169385448456758, 11.72833319551072955152938581395, 12.21487527546427143648477470605, 13.096402659909153645292198599064, 14.65570482297697393036096294119, 16.136221322142946668820806674218, 17.026957867608233042319041511113, 18.10035229843927305014668984656, 18.82178975745682898306417088448, 19.83898529462474806466253360894, 20.66914868722347446214601506888, 22.08047677149673690226567060513, 22.7889058978076153394886914493, 23.5190208966736030717846695857, 24.68993099276760627238517162360, 26.061467909523699261379053377190, 27.37589884226006928968075711963, 27.66839484607074519406763638923

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