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Properties

Label 1-431-431.256-r0-0-0
Degree $1$
Conductor $431$
Sign $0.998 + 0.0563i$
Analytic cond. $2.00155$
Root an. cond. $2.00155$
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  + (−0.976 + 0.217i)2-s + (0.957 + 0.288i)3-s + (0.905 − 0.424i)4-s + (0.391 − 0.920i)5-s + (−0.997 − 0.0729i)6-s + (0.833 + 0.551i)7-s + (−0.791 + 0.611i)8-s + (0.833 + 0.551i)9-s + (−0.181 + 0.983i)10-s + (0.744 − 0.667i)11-s + (0.989 − 0.145i)12-s + (−0.976 − 0.217i)13-s + (−0.934 − 0.357i)14-s + (0.639 − 0.768i)15-s + (0.639 − 0.768i)16-s + (0.744 − 0.667i)17-s + ⋯
L(s)  = 1  + (−0.976 + 0.217i)2-s + (0.957 + 0.288i)3-s + (0.905 − 0.424i)4-s + (0.391 − 0.920i)5-s + (−0.997 − 0.0729i)6-s + (0.833 + 0.551i)7-s + (−0.791 + 0.611i)8-s + (0.833 + 0.551i)9-s + (−0.181 + 0.983i)10-s + (0.744 − 0.667i)11-s + (0.989 − 0.145i)12-s + (−0.976 − 0.217i)13-s + (−0.934 − 0.357i)14-s + (0.639 − 0.768i)15-s + (0.639 − 0.768i)16-s + (0.744 − 0.667i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $0.998 + 0.0563i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ 0.998 + 0.0563i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.479983105 + 0.04176742613i\)
\(L(\frac12)\) \(\approx\) \(1.479983105 + 0.04176742613i\)
\(L(1)\) \(\approx\) \(1.153543607 + 0.05587607172i\)
\(L(1)\) \(\approx\) \(1.153543607 + 0.05587607172i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (-0.976 + 0.217i)T \)
3 \( 1 + (0.957 + 0.288i)T \)
5 \( 1 + (0.391 - 0.920i)T \)
7 \( 1 + (0.833 + 0.551i)T \)
11 \( 1 + (0.744 - 0.667i)T \)
13 \( 1 + (-0.976 - 0.217i)T \)
17 \( 1 + (0.744 - 0.667i)T \)
19 \( 1 + (0.391 + 0.920i)T \)
23 \( 1 + (-0.457 + 0.889i)T \)
29 \( 1 + (0.905 + 0.424i)T \)
31 \( 1 + (-0.997 + 0.0729i)T \)
37 \( 1 + (-0.934 - 0.357i)T \)
41 \( 1 + (0.639 - 0.768i)T \)
43 \( 1 + (-0.181 - 0.983i)T \)
47 \( 1 + (-0.322 + 0.946i)T \)
53 \( 1 + (-0.322 - 0.946i)T \)
59 \( 1 + (-0.976 + 0.217i)T \)
61 \( 1 + (0.520 - 0.853i)T \)
67 \( 1 + (-0.934 - 0.357i)T \)
71 \( 1 + (-0.322 + 0.946i)T \)
73 \( 1 + (-0.872 - 0.489i)T \)
79 \( 1 + (0.252 + 0.967i)T \)
83 \( 1 + (0.957 + 0.288i)T \)
89 \( 1 + (-0.0365 - 0.999i)T \)
97 \( 1 + (0.252 - 0.967i)T \)
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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

−24.54262939938353729367713951544, −23.51482140939963470435442790305, −22.048076604991003877283707682970, −21.36456367811090077140060016829, −20.42030927061326304208617723220, −19.70161735341979977873304325828, −19.06134181844993142227715436512, −17.97973943602943399764355610682, −17.59611791682539289377433335919, −16.54655792385231072494880691911, −15.03682495259434096660205962569, −14.70699317110362877004847648847, −13.79418943496963495160054125241, −12.472599183555993683306262518699, −11.60349673891473889489995561449, −10.4202183997661859645610335910, −9.84155512651328136652720453844, −8.89491027930044456014912016633, −7.76697055122082753411370129620, −7.22111871177033252450457747110, −6.38474177138151910950521081848, −4.43046023711281672846433065211, −3.20504644339249024145601789055, −2.218031295412056206691502998454, −1.39942328063972503823282540206, 1.29711050702977772250484108949, 2.10158781171930284689851966426, 3.41008487625458463585825496744, 4.98202596686057694707677746498, 5.74667892660022998660406821436, 7.36951468467799604837428088529, 8.08468567208397429264980555762, 8.92696641093091186398712782168, 9.4838497889820719463070115148, 10.38763168353782842380975424427, 11.742553864768644186970277259967, 12.444233242770183215702583159850, 14.147267311273556652752558524047, 14.35850927211798906672938338447, 15.63350754428926901480844626001, 16.31092946715537636456494189875, 17.20197747394918702420956826805, 18.0733020757524352399666946126, 19.0673638703203794414112633663, 19.78450416711011134313792968377, 20.6198947311204393628832212778, 21.20369548089621226857239459748, 22.048107754045938343070549394845, 23.88713609575426566009813429341, 24.51111868997500168285360386980

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