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Properties

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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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

\(L(\frac{1}{2})\) \(\approx\) \(3.177393410 - 2.475676824i\)
\(L(\frac12)\) \(\approx\) \(3.177393410 - 2.475676824i\)
\(L(1)\) \(\approx\) \(1.912028572 - 0.7348958482i\)
\(L(1)\) \(\approx\) \(1.912028572 - 0.7348958482i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (0.989 - 0.145i)T \)
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 \( 1 + (0.744 - 0.667i)T \)
29 \( 1 + (0.957 + 0.288i)T \)
31 \( 1 + (0.457 - 0.889i)T \)
37 \( 1 + (0.694 - 0.719i)T \)
41 \( 1 + (0.833 - 0.551i)T \)
43 \( 1 + (-0.391 + 0.920i)T \)
47 \( 1 + (0.976 + 0.217i)T \)
53 \( 1 + (-0.976 + 0.217i)T \)
59 \( 1 + (0.989 - 0.145i)T \)
61 \( 1 + (-0.934 - 0.357i)T \)
67 \( 1 + (0.694 - 0.719i)T \)
71 \( 1 + (0.976 + 0.217i)T \)
73 \( 1 + (0.181 + 0.983i)T \)
79 \( 1 + (-0.639 - 0.768i)T \)
83 \( 1 + (0.322 + 0.946i)T \)
89 \( 1 + (-0.520 - 0.853i)T \)
97 \( 1 + (0.639 - 0.768i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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