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Properties

Label 1-431-431.321-r1-0-0
Degree $1$
Conductor $431$
Sign $-0.994 - 0.102i$
Analytic cond. $46.3173$
Root an. cond. $46.3173$
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  + (−0.791 + 0.611i)2-s + (0.639 + 0.768i)3-s + (0.252 − 0.967i)4-s + (−0.934 + 0.357i)5-s + (−0.976 − 0.217i)6-s + (0.181 − 0.983i)7-s + (0.391 + 0.920i)8-s + (−0.181 + 0.983i)9-s + (0.520 − 0.853i)10-s + (−0.581 − 0.813i)11-s + (0.905 − 0.424i)12-s + (0.791 + 0.611i)13-s + (0.457 + 0.889i)14-s + (−0.872 − 0.489i)15-s + (−0.872 − 0.489i)16-s + (0.581 + 0.813i)17-s + ⋯
L(s)  = 1  + (−0.791 + 0.611i)2-s + (0.639 + 0.768i)3-s + (0.252 − 0.967i)4-s + (−0.934 + 0.357i)5-s + (−0.976 − 0.217i)6-s + (0.181 − 0.983i)7-s + (0.391 + 0.920i)8-s + (−0.181 + 0.983i)9-s + (0.520 − 0.853i)10-s + (−0.581 − 0.813i)11-s + (0.905 − 0.424i)12-s + (0.791 + 0.611i)13-s + (0.457 + 0.889i)14-s + (−0.872 − 0.489i)15-s + (−0.872 − 0.489i)16-s + (0.581 + 0.813i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $-0.994 - 0.102i$
Analytic conductor: \(46.3173\)
Root analytic conductor: \(46.3173\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (1:\ ),\ -0.994 - 0.102i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03797723673 + 0.7360573620i\)
\(L(\frac12)\) \(\approx\) \(-0.03797723673 + 0.7360573620i\)
\(L(1)\) \(\approx\) \(0.6147406666 + 0.3822929351i\)
\(L(1)\) \(\approx\) \(0.6147406666 + 0.3822929351i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (-0.791 + 0.611i)T \)
3 \( 1 + (0.639 + 0.768i)T \)
5 \( 1 + (-0.934 + 0.357i)T \)
7 \( 1 + (0.181 - 0.983i)T \)
11 \( 1 + (-0.581 - 0.813i)T \)
13 \( 1 + (0.791 + 0.611i)T \)
17 \( 1 + (0.581 + 0.813i)T \)
19 \( 1 + (-0.934 - 0.357i)T \)
23 \( 1 + (0.989 - 0.145i)T \)
29 \( 1 + (0.252 + 0.967i)T \)
31 \( 1 + (0.976 - 0.217i)T \)
37 \( 1 + (0.457 + 0.889i)T \)
41 \( 1 + (-0.872 - 0.489i)T \)
43 \( 1 + (-0.520 - 0.853i)T \)
47 \( 1 + (-0.833 + 0.551i)T \)
53 \( 1 + (0.833 + 0.551i)T \)
59 \( 1 + (-0.791 + 0.611i)T \)
61 \( 1 + (-0.997 - 0.0729i)T \)
67 \( 1 + (0.457 + 0.889i)T \)
71 \( 1 + (-0.833 + 0.551i)T \)
73 \( 1 + (0.0365 + 0.999i)T \)
79 \( 1 + (0.694 + 0.719i)T \)
83 \( 1 + (-0.639 - 0.768i)T \)
89 \( 1 + (-0.109 - 0.994i)T \)
97 \( 1 + (-0.694 + 0.719i)T \)
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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

−23.32437514806917414922226626491, −22.88552776021996924358965120872, −21.084223703476151551292584195976, −20.89485635785766689016797063497, −19.8340754084665446435525001367, −19.16562712329683816780674017075, −18.4057910613170249295487813861, −17.87313884029411718474303134460, −16.635298671919292336514947289138, −15.48559676849937643181284537509, −15.0322527080825143108612511483, −13.38536179775621607586131183932, −12.62355049812096299832614371176, −12.02437748854102773892133407567, −11.18730857409425871276185661647, −9.86157412318616852667757695060, −8.86385730673139269856494682388, −8.156335026848982113398012962341, −7.61801166510335389827303130530, −6.42187451134918192542622559624, −4.789178324666440653435225627522, −3.40985311545358959716133050484, −2.609990696268498407535665555282, −1.46427878548844375973443525418, −0.262913141537684962276293345, 1.20815840749251948340458226474, 2.9151956611626613839114720363, 3.981506704128005679793783573842, 4.92887214798077200464641155835, 6.39320388723251028052764222850, 7.376616316707374834991222140643, 8.34721166219066806993796628548, 8.700412282004836934038311771043, 10.22252920232507010646570570733, 10.70356125634962043311254653499, 11.409267756616620969363920094180, 13.304950892424939250327404587285, 14.15619441406934913897619263877, 14.98393672267808756731965950660, 15.666009795827032825484936360391, 16.54434871177002591281569264495, 17.04669473061862621090238950872, 18.60993858353261144406184202693, 19.10573212993993336320688289013, 19.896166921139755606969278759701, 20.73180067361981473672629768185, 21.59155811655195449656330065087, 23.044154558732655278305992826245, 23.56843449300585105400363794463, 24.31297677790138857018078674333

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