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Properties

Label 1-431-431.162-r0-0-0
Degree $1$
Conductor $431$
Sign $-0.999 + 0.00589i$
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  + (−0.872 − 0.489i)2-s + (−0.934 − 0.357i)3-s + (0.520 + 0.853i)4-s + (0.109 − 0.994i)5-s + (0.639 + 0.768i)6-s + (0.744 + 0.667i)7-s + (−0.0365 − 0.999i)8-s + (0.744 + 0.667i)9-s + (−0.581 + 0.813i)10-s + (−0.791 − 0.611i)11-s + (−0.181 − 0.983i)12-s + (−0.872 + 0.489i)13-s + (−0.322 − 0.946i)14-s + (−0.457 + 0.889i)15-s + (−0.457 + 0.889i)16-s + (−0.791 − 0.611i)17-s + ⋯
L(s)  = 1  + (−0.872 − 0.489i)2-s + (−0.934 − 0.357i)3-s + (0.520 + 0.853i)4-s + (0.109 − 0.994i)5-s + (0.639 + 0.768i)6-s + (0.744 + 0.667i)7-s + (−0.0365 − 0.999i)8-s + (0.744 + 0.667i)9-s + (−0.581 + 0.813i)10-s + (−0.791 − 0.611i)11-s + (−0.181 − 0.983i)12-s + (−0.872 + 0.489i)13-s + (−0.322 − 0.946i)14-s + (−0.457 + 0.889i)15-s + (−0.457 + 0.889i)16-s + (−0.791 − 0.611i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $-0.999 + 0.00589i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (162, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ -0.999 + 0.00589i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0009400381040 - 0.3190137413i\)
\(L(\frac12)\) \(\approx\) \(0.0009400381040 - 0.3190137413i\)
\(L(1)\) \(\approx\) \(0.4173114790 - 0.2356625148i\)
\(L(1)\) \(\approx\) \(0.4173114790 - 0.2356625148i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (-0.872 - 0.489i)T \)
3 \( 1 + (-0.934 - 0.357i)T \)
5 \( 1 + (0.109 - 0.994i)T \)
7 \( 1 + (0.744 + 0.667i)T \)
11 \( 1 + (-0.791 - 0.611i)T \)
13 \( 1 + (-0.872 + 0.489i)T \)
17 \( 1 + (-0.791 - 0.611i)T \)
19 \( 1 + (0.109 + 0.994i)T \)
23 \( 1 + (0.833 - 0.551i)T \)
29 \( 1 + (0.520 - 0.853i)T \)
31 \( 1 + (0.639 - 0.768i)T \)
37 \( 1 + (-0.322 - 0.946i)T \)
41 \( 1 + (-0.457 + 0.889i)T \)
43 \( 1 + (-0.581 - 0.813i)T \)
47 \( 1 + (-0.694 - 0.719i)T \)
53 \( 1 + (-0.694 + 0.719i)T \)
59 \( 1 + (-0.872 - 0.489i)T \)
61 \( 1 + (0.957 + 0.288i)T \)
67 \( 1 + (-0.322 - 0.946i)T \)
71 \( 1 + (-0.694 - 0.719i)T \)
73 \( 1 + (0.989 - 0.145i)T \)
79 \( 1 + (-0.997 - 0.0729i)T \)
83 \( 1 + (-0.934 - 0.357i)T \)
89 \( 1 + (0.905 - 0.424i)T \)
97 \( 1 + (-0.997 + 0.0729i)T \)
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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

−24.447354578710958188259505245219, −23.685565809937859969303176754127, −23.079166491458031278710043881805, −22.087395481378139322828603806848, −21.16576369825554355850879362527, −20.12896553199997568099807287474, −19.203827015589005565436239603432, −18.04168634217025320879093252520, −17.609253641670732072319153828933, −17.16144175705585125830000924736, −15.78527625163266548632517857780, −15.22675057091472792588011114039, −14.4704325862708738148238084642, −13.15708104786539710723205438028, −11.721452980911399609054406904137, −10.87042856738968415884810482272, −10.420872783973945985455971926728, −9.63199034644042471356035703123, −8.19119201405247109417234872534, −7.07149630615627577984452631380, −6.77007302025186363374653099661, −5.31619877581473184760562559164, −4.67835028620637120075435295454, −2.85083885509833810502171268551, −1.45382464508578159516946273103, 0.27977493611207441257102226425, 1.60893766502200940509053338062, 2.5050335848920372408146054925, 4.44228407984770924347516597423, 5.24477424818866742832261487600, 6.38463407767659673388449947545, 7.66387951823030166536330938401, 8.35053632058140160198151811444, 9.359571986608638685008383418642, 10.36106129439554453943438498586, 11.41443446839838886978784794443, 11.95862033038009636373308847905, 12.73507396361734170534478623196, 13.66165280540409854477084929623, 15.3765538797528006646754879375, 16.24250153710251952680141081962, 16.94250623078948173713681343568, 17.636595933220612945431925311515, 18.513656749089959185702575381369, 19.09102438635284609301897192156, 20.29606763817764308307022632990, 21.21289633961331778486898593830, 21.60893070446039853532342561401, 22.77544242305568511470290577199, 24.00365435544111706828267962929

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