L-function 1-431-431.282-r1-0-0

• Label: 1-431-431.282-r1-0-0
• Degree: $1$
• Conductor: $431$
• Sign: $0.244 - 0.969i$
• 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

+ (0.989 − 0.145i)2^-s + (−0.322 − 0.946i)3^-s + (0.957 − 0.288i)4^-s + (0.252 + 0.967i)5^-s + (−0.457 − 0.889i)6^-s + (0.791 − 0.611i)7^-s + (0.905 − 0.424i)8^-s + (−0.791 + 0.611i)9^-s + (0.391 + 0.920i)10^-s + (−0.0365 + 0.999i)11^-s + (−0.581 − 0.813i)12^-s + (−0.989 − 0.145i)13^-s + (0.694 − 0.719i)14^-s + (0.833 − 0.551i)15^-s + (0.833 − 0.551i)16^-s + (0.0365 − 0.999i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$431$
Sign:	$0.244 - 0.969i$
Analytic conductor:	$46.3173$
Root analytic conductor:	$46.3173$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{431} (282, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 431,\ (1:\ ),\ 0.244 - 0.969i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$3.177393410 - 2.475676824i$
$L(1)$	$\approx$	$1.912028572 - 0.7348958482i$

Euler product

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

	$p$	$F_p(T)$
bad	431	$1$
good	2	$1 + (0.989 - 0.145i)T$
	3	$1 + (-0.322 - 0.946i)T$
	5	$1 + (0.252 + 0.967i)T$
	7	$1 + (0.791 - 0.611i)T$
	11	$1 + (-0.0365 + 0.999i)T$
	13	$1 + (-0.989 - 0.145i)T$
	17	$1 + (0.0365 - 0.999i)T$
	19	$1 + (0.252 - 0.967i)T$
	23	$1 + (0.744 - 0.667i)T$

Imaginary part of the first few zeros on the critical line

−23.984179876026464142498646031412, −23.336466746251627826503242100483, −22.08692544053333041006607865403, −21.46657658619519435567890281302, −21.15306958760064604600350414762, −20.184473225225114571566670574389, −19.22448588386620451552444889635, −17.53180735341430444772910855241, −16.937582735287907126165854280096, −16.1658158994533665898346420713, −15.32424122677249711209522332445, −14.528109875989246159703606476829, −13.71289420231345776371833532007, −12.42672016964083011886982145877, −11.889863867931706053249976906376, −10.96232402277224108423770585670, −9.92481045100883320892845999464, −8.695685370493932086938581208247, −7.97732107006927435630616627256, −6.24116801197626357428616062428, −5.455682684545610825375443126203, −4.87213015471425529201828336513, −3.92475131269674205612057196088, −2.71083513142933326884895113427, −1.287633497631727851931375205213, 0.84817129631007669367416926581, 2.25570947331314886008943784710, 2.76039474204112951943918722572, 4.50243108291000789892938443928, 5.21283864579809695734406943878, 6.51936156395605575756503694340, 7.20719507767873875072203515997, 7.6956972898117976373039324667, 9.70356126349998188313698711846, 10.79600035030334887303293990393, 11.39270391596863464921677857236, 12.300588448205183317495610642168, 13.19546210529822956013958634006, 14.15106778369077420445271687461, 14.54298370101945026854974454133, 15.58507581066875149125478259536, 17.01096847335201620823815114240, 17.65827384224092838693994955331, 18.53354007093143815528067065315, 19.62147015169093177046122251209, 20.252363673007366549681650694450, 21.31429066894113110932394178876, 22.35180572027238338886559790624, 22.86805452583424264209674664570, 23.55866741510626498775875917131

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