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Properties

Label 1-431-431.132-r0-0-0
Degree $1$
Conductor $431$
Sign $-0.276 + 0.961i$
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 + (0.782 + 0.622i)5-s + (−0.0365 + 0.999i)6-s + (−0.605 − 0.795i)7-s + (−0.322 − 0.946i)8-s + (0.957 + 0.288i)9-s + (−0.533 + 0.845i)10-s + (0.0511 + 0.998i)11-s + (−0.997 + 0.0729i)12-s + (0.979 − 0.203i)13-s + (0.724 − 0.688i)14-s + (0.683 + 0.729i)15-s + (0.905 − 0.424i)16-s + (0.965 + 0.259i)17-s + ⋯
L(s)  = 1  + (0.109 + 0.994i)2-s + (0.989 + 0.145i)3-s + (−0.976 + 0.217i)4-s + (0.782 + 0.622i)5-s + (−0.0365 + 0.999i)6-s + (−0.605 − 0.795i)7-s + (−0.322 − 0.946i)8-s + (0.957 + 0.288i)9-s + (−0.533 + 0.845i)10-s + (0.0511 + 0.998i)11-s + (−0.997 + 0.0729i)12-s + (0.979 − 0.203i)13-s + (0.724 − 0.688i)14-s + (0.683 + 0.729i)15-s + (0.905 − 0.424i)16-s + (0.965 + 0.259i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $-0.276 + 0.961i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ -0.276 + 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.201520371 + 1.595499190i\)
\(L(\frac12)\) \(\approx\) \(1.201520371 + 1.595499190i\)
\(L(1)\) \(\approx\) \(1.241294125 + 0.9156466731i\)
\(L(1)\) \(\approx\) \(1.241294125 + 0.9156466731i\)

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 \( 1 + (0.782 + 0.622i)T \)
7 \( 1 + (-0.605 - 0.795i)T \)
11 \( 1 + (0.0511 + 0.998i)T \)
13 \( 1 + (0.979 - 0.203i)T \)
17 \( 1 + (0.965 + 0.259i)T \)
19 \( 1 + (-0.999 + 0.0438i)T \)
23 \( 1 + (0.0802 - 0.996i)T \)
29 \( 1 + (-0.508 + 0.861i)T \)
31 \( 1 + (0.616 + 0.787i)T \)
37 \( 1 + (-0.991 + 0.131i)T \)
41 \( 1 + (-0.483 + 0.875i)T \)
43 \( 1 + (-0.533 - 0.845i)T \)
47 \( 1 + (-0.581 - 0.813i)T \)
53 \( 1 + (0.948 - 0.315i)T \)
59 \( 1 + (-0.672 - 0.739i)T \)
61 \( 1 + (0.417 - 0.908i)T \)
67 \( 1 + (0.879 + 0.476i)T \)
71 \( 1 + (0.593 - 0.804i)T \)
73 \( 1 + (-0.773 + 0.634i)T \)
79 \( 1 + (-0.825 + 0.563i)T \)
83 \( 1 + (0.444 - 0.895i)T \)
89 \( 1 + (0.138 - 0.990i)T \)
97 \( 1 + (0.336 + 0.941i)T \)
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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

−23.94784590777641022148405227288, −22.86310225225061885757047918718, −21.66103933653227372654079802005, −21.20737645005194702948158738547, −20.6785886727231249665640345870, −19.45305743213313783880583308669, −18.96640174060340144238715583204, −18.266043593927696981092013519182, −17.06601124564627379426563623383, −15.961345194585386644842926520027, −14.88795172619992297326728574897, −13.73098786913283740711487191890, −13.417977469173897310335065831741, −12.553131494596785503347209162598, −11.60360010544086383108153968611, −10.285050877353557237904913424, −9.45154920070941356960780573253, −8.82480880910897576443890772822, −8.1348741885660326190755998519, −6.240936825973948470019852525823, −5.45897417260100905548336925122, −4.01329013740686336003396303353, −3.13409918595047908912209219213, −2.1705288124827816152257528910, −1.15013606282797432874813972536, 1.59934708993909840890777170067, 3.15592982005803965194863175882, 3.90473158609320002765038554673, 5.1081092107090604023147289258, 6.550350436681631855378714497223, 6.93523214874903772049255976308, 8.1020471691605363431134641553, 8.98360592720492126847769783779, 10.1075715317782790081584747457, 10.369334084787664585720506515856, 12.63540410837719582176773923288, 13.209030382961023270196684097834, 14.09661730150930186968069734120, 14.67161869021803559436599811235, 15.48964282292277270080781178212, 16.51157823798942052640239427759, 17.304548048537861569424838384210, 18.380813834202812947197917848089, 18.94814603111767661247148091234, 20.155083781649976136583813253261, 21.03892417827598471293392906594, 21.86989458948484666493598263777, 22.94541259800050685980142245632, 23.38171080151887884886579200103, 24.718704920494942727105357849272

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