L-function 1-431-431.321-r1-0-0

• Label: 1-431-431.321-r1-0-0
• Degree: $1$
• Conductor: $431$
• Sign: $-0.994 - 0.102i$
• Analytic cond.: $46.3173$
• Root an. cond.: $46.3173$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.791 + 0.611i)2^-s + (0.639 + 0.768i)3^-s + (0.252 − 0.967i)4^-s + (−0.934 + 0.357i)5^-s + (−0.976 − 0.217i)6^-s + (0.181 − 0.983i)7^-s + (0.391 + 0.920i)8^-s + (−0.181 + 0.983i)9^-s + (0.520 − 0.853i)10^-s + (−0.581 − 0.813i)11^-s + (0.905 − 0.424i)12^-s + (0.791 + 0.611i)13^-s + (0.457 + 0.889i)14^-s + (−0.872 − 0.489i)15^-s + (−0.872 − 0.489i)16^-s + (0.581 + 0.813i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$431$
Sign:	$-0.994 - 0.102i$
Analytic conductor:	$46.3173$
Root analytic conductor:	$46.3173$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{431} (321, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 431,\ (1:\ ),\ -0.994 - 0.102i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-0.03797723673 + 0.7360573620i$
$L(1)$	$\approx$	$0.6147406666 + 0.3822929351i$

Euler product

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

	$p$	$F_p(T)$
bad	431	$1$
good	2	$1 + (-0.791 + 0.611i)T$
	3	$1 + (0.639 + 0.768i)T$
	5	$1 + (-0.934 + 0.357i)T$
	7	$1 + (0.181 - 0.983i)T$
	11	$1 + (-0.581 - 0.813i)T$
	13	$1 + (0.791 + 0.611i)T$
	17	$1 + (0.581 + 0.813i)T$
	19	$1 + (-0.934 - 0.357i)T$
	23	$1 + (0.989 - 0.145i)T$

Imaginary part of the first few zeros on the critical line

−23.32437514806917414922226626491, −22.88552776021996924358965120872, −21.084223703476151551292584195976, −20.89485635785766689016797063497, −19.8340754084665446435525001367, −19.16562712329683816780674017075, −18.4057910613170249295487813861, −17.87313884029411718474303134460, −16.635298671919292336514947289138, −15.48559676849937643181284537509, −15.0322527080825143108612511483, −13.38536179775621607586131183932, −12.62355049812096299832614371176, −12.02437748854102773892133407567, −11.18730857409425871276185661647, −9.86157412318616852667757695060, −8.86385730673139269856494682388, −8.156335026848982113398012962341, −7.61801166510335389827303130530, −6.42187451134918192542622559624, −4.789178324666440653435225627522, −3.40985311545358959716133050484, −2.609990696268498407535665555282, −1.46427878548844375973443525418, −0.262913141537684962276293345, 1.20815840749251948340458226474, 2.9151956611626613839114720363, 3.981506704128005679793783573842, 4.92887214798077200464641155835, 6.39320388723251028052764222850, 7.376616316707374834991222140643, 8.34721166219066806993796628548, 8.700412282004836934038311771043, 10.22252920232507010646570570733, 10.70356125634962043311254653499, 11.409267756616620969363920094180, 13.304950892424939250327404587285, 14.15619441406934913897619263877, 14.98393672267808756731965950660, 15.666009795827032825484936360391, 16.54434871177002591281569264495, 17.04669473061862621090238950872, 18.60993858353261144406184202693, 19.10573212993993336320688289013, 19.896166921139755606969278759701, 20.73180067361981473672629768185, 21.59155811655195449656330065087, 23.044154558732655278305992826245, 23.56843449300585105400363794463, 24.31297677790138857018078674333

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