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Properties

Label 1-431-431.17-r1-0-0
Degree $1$
Conductor $431$
Sign $0.879 + 0.476i$
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  + (0.639 − 0.768i)2-s + (0.391 + 0.920i)3-s + (−0.181 − 0.983i)4-s + (−0.557 − 0.829i)5-s + (0.957 + 0.288i)6-s + (0.899 + 0.437i)7-s + (−0.872 − 0.489i)8-s + (−0.694 + 0.719i)9-s + (−0.994 − 0.102i)10-s + (0.661 + 0.749i)11-s + (0.833 − 0.551i)12-s + (0.969 + 0.245i)13-s + (0.911 − 0.411i)14-s + (0.545 − 0.838i)15-s + (−0.934 + 0.357i)16-s + (0.0948 + 0.995i)17-s + ⋯
L(s)  = 1  + (0.639 − 0.768i)2-s + (0.391 + 0.920i)3-s + (−0.181 − 0.983i)4-s + (−0.557 − 0.829i)5-s + (0.957 + 0.288i)6-s + (0.899 + 0.437i)7-s + (−0.872 − 0.489i)8-s + (−0.694 + 0.719i)9-s + (−0.994 − 0.102i)10-s + (0.661 + 0.749i)11-s + (0.833 − 0.551i)12-s + (0.969 + 0.245i)13-s + (0.911 − 0.411i)14-s + (0.545 − 0.838i)15-s + (−0.934 + 0.357i)16-s + (0.0948 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $0.879 + 0.476i$
Analytic conductor: \(46.3173\)
Root analytic conductor: \(46.3173\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (1:\ ),\ 0.879 + 0.476i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.798631298 + 0.7090345291i\)
\(L(\frac12)\) \(\approx\) \(2.798631298 + 0.7090345291i\)
\(L(1)\) \(\approx\) \(1.619135798 - 0.1323357355i\)
\(L(1)\) \(\approx\) \(1.619135798 - 0.1323357355i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (0.639 - 0.768i)T \)
3 \( 1 + (0.391 + 0.920i)T \)
5 \( 1 + (-0.557 - 0.829i)T \)
7 \( 1 + (0.899 + 0.437i)T \)
11 \( 1 + (0.661 + 0.749i)T \)
13 \( 1 + (0.969 + 0.245i)T \)
17 \( 1 + (0.0948 + 0.995i)T \)
19 \( 1 + (-0.961 - 0.274i)T \)
23 \( 1 + (-0.999 - 0.0146i)T \)
29 \( 1 + (0.724 - 0.688i)T \)
31 \( 1 + (-0.0219 + 0.999i)T \)
37 \( 1 + (0.672 + 0.739i)T \)
41 \( 1 + (0.0511 + 0.998i)T \)
43 \( 1 + (0.994 - 0.102i)T \)
47 \( 1 + (-0.252 - 0.967i)T \)
53 \( 1 + (0.998 - 0.0584i)T \)
59 \( 1 + (-0.533 - 0.845i)T \)
61 \( 1 + (0.593 + 0.804i)T \)
67 \( 1 + (-0.495 + 0.868i)T \)
71 \( 1 + (0.773 + 0.634i)T \)
73 \( 1 + (0.987 + 0.160i)T \)
79 \( 1 + (0.923 - 0.384i)T \)
83 \( 1 + (-0.224 + 0.974i)T \)
89 \( 1 + (-0.167 + 0.985i)T \)
97 \( 1 + (0.0802 - 0.996i)T \)
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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

−23.914268534581237987946747844361, −23.1725973646951561774688830, −22.51622461609144635804345464861, −21.344455915827102806376041069769, −20.48260305359867230260465735870, −19.49478230694955604511970666361, −18.382447391119005844470835108064, −17.94767522441285908588604881846, −16.86712883034559320737802641003, −15.83874252015433072566946846559, −14.824710747084233184741288033073, −14.09616755165767684545614781060, −13.74339969338792398444182239642, −12.46207003202979341816550705322, −11.586964452674417773095387605862, −10.89948122740052286078119494172, −8.96742652624244464209481763956, −8.093077634090396983635150736524, −7.53059894248923439219475501065, −6.536813256823261151852135248403, −5.843963446690305763176487331700, −4.21403140378928111398434056253, −3.46585145684805437982137744274, −2.293614439314262317367711852320, −0.62213250621931615964035815691, 1.29234653334718502347563009464, 2.29939643547685540808715008326, 3.88118426321039265227179394785, 4.26384028228498494323457191535, 5.15268406911227748536626749254, 6.264543123713371043904732552752, 8.22313514237386375710282787220, 8.728621660993741369723868560737, 9.760457938258097964903780236017, 10.77875152649428482976726808755, 11.59937275853873504607250706494, 12.33355520686737590373197940879, 13.42352275615940728253177446298, 14.44162296918441604221694613851, 15.12044452583558613218676228317, 15.789868446858453874659833717072, 16.94149665199537242092890831797, 18.0060584111051060773988514163, 19.33262814380233115099664802603, 19.87845638944486017620777713322, 20.71155144153688428325316674258, 21.30657042573981031290011294928, 21.944649738981312396864219191407, 23.14363144739848130679312226636, 23.712220972741518274577263826667

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