L-function 1-431-431.17-r1-0-0

• Label: 1-431-431.17-r1-0-0
• Degree: $1$
• Conductor: $431$
• Sign: $0.879 + 0.476i$
• 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

+ (0.639 − 0.768i)2^-s + (0.391 + 0.920i)3^-s + (−0.181 − 0.983i)4^-s + (−0.557 − 0.829i)5^-s + (0.957 + 0.288i)6^-s + (0.899 + 0.437i)7^-s + (−0.872 − 0.489i)8^-s + (−0.694 + 0.719i)9^-s + (−0.994 − 0.102i)10^-s + (0.661 + 0.749i)11^-s + (0.833 − 0.551i)12^-s + (0.969 + 0.245i)13^-s + (0.911 − 0.411i)14^-s + (0.545 − 0.838i)15^-s + (−0.934 + 0.357i)16^-s + (0.0948 + 0.995i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$431$
Sign:	$0.879 + 0.476i$
Analytic conductor:	$46.3173$
Root analytic conductor:	$46.3173$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{431} (17, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 431,\ (1:\ ),\ 0.879 + 0.476i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.798631298 + 0.7090345291i$
$L(1)$	$\approx$	$1.619135798 - 0.1323357355i$

Euler product

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

	$p$	$F_p(T)$
bad	431	$1$
good	2	$1 + (0.639 - 0.768i)T$
	3	$1 + (0.391 + 0.920i)T$
	5	$1 + (-0.557 - 0.829i)T$
	7	$1 + (0.899 + 0.437i)T$
	11	$1 + (0.661 + 0.749i)T$
	13	$1 + (0.969 + 0.245i)T$
	17	$1 + (0.0948 + 0.995i)T$
	19	$1 + (-0.961 - 0.274i)T$
	23	$1 + (-0.999 - 0.0146i)T$

Imaginary part of the first few zeros on the critical line

−23.914268534581237987946747844361, −23.1725973646951561774688830, −22.51622461609144635804345464861, −21.344455915827102806376041069769, −20.48260305359867230260465735870, −19.49478230694955604511970666361, −18.382447391119005844470835108064, −17.94767522441285908588604881846, −16.86712883034559320737802641003, −15.83874252015433072566946846559, −14.824710747084233184741288033073, −14.09616755165767684545614781060, −13.74339969338792398444182239642, −12.46207003202979341816550705322, −11.586964452674417773095387605862, −10.89948122740052286078119494172, −8.96742652624244464209481763956, −8.093077634090396983635150736524, −7.53059894248923439219475501065, −6.536813256823261151852135248403, −5.843963446690305763176487331700, −4.21403140378928111398434056253, −3.46585145684805437982137744274, −2.293614439314262317367711852320, −0.62213250621931615964035815691, 1.29234653334718502347563009464, 2.29939643547685540808715008326, 3.88118426321039265227179394785, 4.26384028228498494323457191535, 5.15268406911227748536626749254, 6.264543123713371043904732552752, 8.22313514237386375710282787220, 8.728621660993741369723868560737, 9.760457938258097964903780236017, 10.77875152649428482976726808755, 11.59937275853873504607250706494, 12.33355520686737590373197940879, 13.42352275615940728253177446298, 14.44162296918441604221694613851, 15.12044452583558613218676228317, 15.789868446858453874659833717072, 16.94149665199537242092890831797, 18.0060584111051060773988514163, 19.33262814380233115099664802603, 19.87845638944486017620777713322, 20.71155144153688428325316674258, 21.30657042573981031290011294928, 21.944649738981312396864219191407, 23.14363144739848130679312226636, 23.712220972741518274577263826667

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