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Properties

Label 1-431-431.16-r0-0-0
Degree $1$
Conductor $431$
Sign $0.307 + 0.951i$
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  + (0.109 + 0.994i)2-s + (0.989 + 0.145i)3-s + (−0.976 + 0.217i)4-s + (0.833 − 0.551i)5-s + (−0.0365 + 0.999i)6-s + (0.957 + 0.288i)7-s + (−0.322 − 0.946i)8-s + (0.957 + 0.288i)9-s + (0.639 + 0.768i)10-s + (−0.934 + 0.357i)11-s + (−0.997 + 0.0729i)12-s + (0.109 − 0.994i)13-s + (−0.181 + 0.983i)14-s + (0.905 − 0.424i)15-s + (0.905 − 0.424i)16-s + (−0.934 + 0.357i)17-s + ⋯
L(s)  = 1  + (0.109 + 0.994i)2-s + (0.989 + 0.145i)3-s + (−0.976 + 0.217i)4-s + (0.833 − 0.551i)5-s + (−0.0365 + 0.999i)6-s + (0.957 + 0.288i)7-s + (−0.322 − 0.946i)8-s + (0.957 + 0.288i)9-s + (0.639 + 0.768i)10-s + (−0.934 + 0.357i)11-s + (−0.997 + 0.0729i)12-s + (0.109 − 0.994i)13-s + (−0.181 + 0.983i)14-s + (0.905 − 0.424i)15-s + (0.905 − 0.424i)16-s + (−0.934 + 0.357i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $0.307 + 0.951i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ 0.307 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.771920076 + 1.288904936i\)
\(L(\frac12)\) \(\approx\) \(1.771920076 + 1.288904936i\)
\(L(1)\) \(\approx\) \(1.474823010 + 0.7631063531i\)
\(L(1)\) \(\approx\) \(1.474823010 + 0.7631063531i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (0.109 + 0.994i)T \)
3 \( 1 + (0.989 + 0.145i)T \)
5 \( 1 + (0.833 - 0.551i)T \)
7 \( 1 + (0.957 + 0.288i)T \)
11 \( 1 + (-0.934 + 0.357i)T \)
13 \( 1 + (0.109 - 0.994i)T \)
17 \( 1 + (-0.934 + 0.357i)T \)
19 \( 1 + (0.833 + 0.551i)T \)
23 \( 1 + (0.520 + 0.853i)T \)
29 \( 1 + (-0.976 - 0.217i)T \)
31 \( 1 + (-0.0365 - 0.999i)T \)
37 \( 1 + (-0.181 + 0.983i)T \)
41 \( 1 + (0.905 - 0.424i)T \)
43 \( 1 + (0.639 - 0.768i)T \)
47 \( 1 + (-0.581 - 0.813i)T \)
53 \( 1 + (-0.581 + 0.813i)T \)
59 \( 1 + (0.109 + 0.994i)T \)
61 \( 1 + (-0.872 + 0.489i)T \)
67 \( 1 + (-0.181 + 0.983i)T \)
71 \( 1 + (-0.581 - 0.813i)T \)
73 \( 1 + (0.252 - 0.967i)T \)
79 \( 1 + (-0.791 - 0.611i)T \)
83 \( 1 + (0.989 + 0.145i)T \)
89 \( 1 + (-0.694 + 0.719i)T \)
97 \( 1 + (-0.791 + 0.611i)T \)
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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

−24.11996545164993837262558735715, −22.95442080181290474322048425365, −21.83452683511261308986040541585, −21.19156749417775221006960831121, −20.70681948712511577486877660663, −19.78536266454732351657849394493, −18.76047472737517240801635062571, −18.19357742587953370192530988257, −17.53458457470406089307167587133, −15.987221904900065680893784098706, −14.63262048343093847499183018352, −14.1882782764210744399185745419, −13.46538613819472482501717738173, −12.72736283293894532587770987579, −11.19552212716606919575708387784, −10.79374602093552381668226656910, −9.54779941300454495002600818010, −8.97115058179242085519428619612, −7.881873636881030242239909866011, −6.782025968819781478954421158677, −5.23494405454810916527021009225, −4.34067502767259820439571587943, −3.03743935010566902863267484686, −2.295731272617523324136006152290, −1.39534887038816139253446134304, 1.49101165572998679452787910137, 2.72933765946173960702291365131, 4.15518056982305341213861606282, 5.13985641052557000222280542625, 5.78536821074071260485070750325, 7.40827205460926971417201416320, 7.99638654146519084278662277700, 8.861641527899860761407458373768, 9.633448888603089688696262008065, 10.61744084783445888408954509982, 12.3874194322777691627045690609, 13.373686966154896453784814365724, 13.652379144219857222233311145809, 14.987140404941520123746558547018, 15.251406844802352161681155910149, 16.31918961235799209618246313720, 17.49659354400571388691666912136, 17.99778213394476843642230509424, 18.86918407749784759835706736872, 20.35882450575623134326023488567, 20.823629015202962162280968336581, 21.69944314381279203213895209014, 22.561077521338284050464849045886, 23.93476316619473643620450013931, 24.438180995295318318504482484178

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