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Properties

Label 1-431-431.32-r0-0-0
Degree $1$
Conductor $431$
Sign $0.881 + 0.472i$
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  + (0.252 − 0.967i)2-s + (−0.181 + 0.983i)3-s + (−0.872 − 0.489i)4-s + (0.744 − 0.667i)5-s + (0.905 + 0.424i)6-s + (−0.934 − 0.357i)7-s + (−0.694 + 0.719i)8-s + (−0.934 − 0.357i)9-s + (−0.457 − 0.889i)10-s + (−0.322 + 0.946i)11-s + (0.639 − 0.768i)12-s + (0.252 + 0.967i)13-s + (−0.581 + 0.813i)14-s + (0.520 + 0.853i)15-s + (0.520 + 0.853i)16-s + (−0.322 + 0.946i)17-s + ⋯
L(s)  = 1  + (0.252 − 0.967i)2-s + (−0.181 + 0.983i)3-s + (−0.872 − 0.489i)4-s + (0.744 − 0.667i)5-s + (0.905 + 0.424i)6-s + (−0.934 − 0.357i)7-s + (−0.694 + 0.719i)8-s + (−0.934 − 0.357i)9-s + (−0.457 − 0.889i)10-s + (−0.322 + 0.946i)11-s + (0.639 − 0.768i)12-s + (0.252 + 0.967i)13-s + (−0.581 + 0.813i)14-s + (0.520 + 0.853i)15-s + (0.520 + 0.853i)16-s + (−0.322 + 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $0.881 + 0.472i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ 0.881 + 0.472i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.015355323 + 0.2551275220i\)
\(L(\frac12)\) \(\approx\) \(1.015355323 + 0.2551275220i\)
\(L(1)\) \(\approx\) \(0.9666886458 - 0.1093188567i\)
\(L(1)\) \(\approx\) \(0.9666886458 - 0.1093188567i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (0.252 - 0.967i)T \)
3 \( 1 + (-0.181 + 0.983i)T \)
5 \( 1 + (0.744 - 0.667i)T \)
7 \( 1 + (-0.934 - 0.357i)T \)
11 \( 1 + (-0.322 + 0.946i)T \)
13 \( 1 + (0.252 + 0.967i)T \)
17 \( 1 + (-0.322 + 0.946i)T \)
19 \( 1 + (0.744 + 0.667i)T \)
23 \( 1 + (0.957 - 0.288i)T \)
29 \( 1 + (-0.872 + 0.489i)T \)
31 \( 1 + (0.905 - 0.424i)T \)
37 \( 1 + (-0.581 + 0.813i)T \)
41 \( 1 + (0.520 + 0.853i)T \)
43 \( 1 + (-0.457 + 0.889i)T \)
47 \( 1 + (0.391 - 0.920i)T \)
53 \( 1 + (0.391 + 0.920i)T \)
59 \( 1 + (0.252 - 0.967i)T \)
61 \( 1 + (0.989 + 0.145i)T \)
67 \( 1 + (-0.581 + 0.813i)T \)
71 \( 1 + (0.391 - 0.920i)T \)
73 \( 1 + (-0.997 + 0.0729i)T \)
79 \( 1 + (-0.0365 + 0.999i)T \)
83 \( 1 + (-0.181 + 0.983i)T \)
89 \( 1 + (-0.976 + 0.217i)T \)
97 \( 1 + (-0.0365 - 0.999i)T \)
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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

−24.34529649747043067507277103539, −23.06683570871360781663955130780, −22.61459736981991396452799178589, −21.9731799517248683589061258282, −20.7924830709460914086818717674, −19.28369112205615596323600635739, −18.69550498946126644520981200404, −17.89962065669921436261180936139, −17.294256892727880445562027271957, −16.15936176765046732193116128051, −15.4194941571489272967702710890, −14.23609319967355551584437000318, −13.34184310832283140501464711257, −13.19527723363821676087099275119, −11.89505252711850712513801248101, −10.732650965851119676371663909078, −9.43530427423229185886223534300, −8.626489403286520119065614658966, −7.35949869597385680672268708729, −6.81041184109576890517626587058, −5.7393100919483494012195203207, −5.42231688734982716751050748477, −3.26366991698823134653882548407, −2.68718449579309449490957343006, −0.60376016993225326587820750136, 1.36048541117032325128713301483, 2.6608013114815104645281963899, 3.8429189969858312647550195823, 4.6002761667480561462507998510, 5.54545822552289543468890249140, 6.54317750258506199588819010029, 8.49097765656397833492146674403, 9.43650704167847300346901528185, 9.87425515398299902763096928291, 10.663677708333014572053635374509, 11.79431109494064624395891356987, 12.73367264163384194551676449379, 13.44820270725998081959175912881, 14.43210884774792238986464030480, 15.4020347820271006462375985956, 16.574953551083412783286949341607, 17.13202411129874256832192606642, 18.18880360565519214759753837341, 19.30114128454855819131633016531, 20.26834153715741927513721214471, 20.7493082032222090055715995478, 21.52608776014622231187161060921, 22.31780858216821419461128346022, 23.027715726561995439022456531173, 23.84880323928086442753772299228

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