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Properties

Label 1-431-431.122-r0-0-0
Degree $1$
Conductor $431$
Sign $-0.895 + 0.445i$
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  + (0.520 + 0.853i)2-s + (0.744 + 0.667i)3-s + (−0.457 + 0.889i)4-s + (0.661 + 0.749i)5-s + (−0.181 + 0.983i)6-s + (0.979 + 0.203i)7-s + (−0.997 + 0.0729i)8-s + (0.109 + 0.994i)9-s + (−0.295 + 0.955i)10-s + (−0.773 − 0.634i)11-s + (−0.934 + 0.357i)12-s + (−0.923 + 0.384i)13-s + (0.336 + 0.941i)14-s + (−0.00730 + 0.999i)15-s + (−0.581 − 0.813i)16-s + (0.998 + 0.0584i)17-s + ⋯
L(s)  = 1  + (0.520 + 0.853i)2-s + (0.744 + 0.667i)3-s + (−0.457 + 0.889i)4-s + (0.661 + 0.749i)5-s + (−0.181 + 0.983i)6-s + (0.979 + 0.203i)7-s + (−0.997 + 0.0729i)8-s + (0.109 + 0.994i)9-s + (−0.295 + 0.955i)10-s + (−0.773 − 0.634i)11-s + (−0.934 + 0.357i)12-s + (−0.923 + 0.384i)13-s + (0.336 + 0.941i)14-s + (−0.00730 + 0.999i)15-s + (−0.581 − 0.813i)16-s + (0.998 + 0.0584i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $-0.895 + 0.445i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ -0.895 + 0.445i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5401428822 + 2.298911326i\)
\(L(\frac12)\) \(\approx\) \(0.5401428822 + 2.298911326i\)
\(L(1)\) \(\approx\) \(1.137862933 + 1.397369184i\)
\(L(1)\) \(\approx\) \(1.137862933 + 1.397369184i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (0.520 + 0.853i)T \)
3 \( 1 + (0.744 + 0.667i)T \)
5 \( 1 + (0.661 + 0.749i)T \)
7 \( 1 + (0.979 + 0.203i)T \)
11 \( 1 + (-0.773 - 0.634i)T \)
13 \( 1 + (-0.923 + 0.384i)T \)
17 \( 1 + (0.998 + 0.0584i)T \)
19 \( 1 + (-0.508 - 0.861i)T \)
23 \( 1 + (-0.754 - 0.656i)T \)
29 \( 1 + (0.892 + 0.450i)T \)
31 \( 1 + (0.879 - 0.476i)T \)
37 \( 1 + (0.999 - 0.0292i)T \)
41 \( 1 + (0.593 - 0.804i)T \)
43 \( 1 + (-0.295 - 0.955i)T \)
47 \( 1 + (-0.0365 + 0.999i)T \)
53 \( 1 + (-0.961 - 0.274i)T \)
59 \( 1 + (-0.650 + 0.759i)T \)
61 \( 1 + (-0.267 + 0.963i)T \)
67 \( 1 + (0.281 - 0.959i)T \)
71 \( 1 + (0.616 - 0.787i)T \)
73 \( 1 + (0.0219 - 0.999i)T \)
79 \( 1 + (-0.885 + 0.463i)T \)
83 \( 1 + (-0.994 - 0.102i)T \)
89 \( 1 + (0.928 + 0.370i)T \)
97 \( 1 + (-0.714 + 0.699i)T \)
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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

−23.547424156914421746880395242997, −23.25907750287676309975922704824, −21.52494454298937573519374032897, −21.18976943316525152257001406507, −20.317126808942296250053181019814, −19.82284810422236845804962259888, −18.663060024192743659697408434466, −17.89934301972939515456234634820, −17.21608402033418269413293236406, −15.56680583339318353191630090620, −14.476444260947707047135383343, −14.11782324903622628929844477948, −13.022308840039545078320701854458, −12.47242353218983561281258670779, −11.67950330738207581055860975568, −10.074794884791936491561755535068, −9.7856422205922258932052491312, −8.34565762002230054926315720411, −7.75670511171539317737064320252, −6.15116051940461560108121021560, −5.12665589727263875133343339555, −4.29845295283645752238373928682, −2.831564239929463982780451986965, −1.96346946373992235227816487361, −1.12188008910161097196785586963, 2.33172445330335711575958683245, 2.97491293764951172849269448212, 4.36767315661171437133486659274, 5.13508831904731312203362737724, 6.1220895618759798068090026998, 7.4340178118169931269760847995, 8.11438170737483514500052122461, 9.077610313304786522992025704797, 10.123525920424962351204570405534, 11.063243017678487285797958224473, 12.33703640967996468978944532662, 13.62331821030398474585078200638, 14.14994100726900915016013386449, 14.78900598911436365006678132352, 15.507918335080012766361925987693, 16.54670808181548788056277938831, 17.402886280144800311173764845360, 18.33122752113111832385364194170, 19.18618114487120157338637964077, 20.639803023598101159542884148444, 21.460973096042576364154170930059, 21.674152047717319004461400628664, 22.65396210448817293957872893958, 23.90040619515504828953450361323, 24.47134006230542885226642895439

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