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Properties

Label 1-23e2-529.131-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.511 - 0.859i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
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  + (−0.263 + 0.964i)2-s + (−0.743 + 0.668i)3-s + (−0.860 − 0.508i)4-s + (−0.239 − 0.970i)5-s + (−0.449 − 0.893i)6-s + (0.179 − 0.983i)7-s + (0.717 − 0.696i)8-s + (0.105 − 0.994i)9-s + (0.999 + 0.0248i)10-s + (−0.944 + 0.329i)11-s + (0.980 − 0.197i)12-s + (0.437 + 0.899i)13-s + (0.901 + 0.432i)14-s + (0.827 + 0.561i)15-s + (0.481 + 0.876i)16-s + (−0.673 − 0.739i)17-s + ⋯
L(s)  = 1  + (−0.263 + 0.964i)2-s + (−0.743 + 0.668i)3-s + (−0.860 − 0.508i)4-s + (−0.239 − 0.970i)5-s + (−0.449 − 0.893i)6-s + (0.179 − 0.983i)7-s + (0.717 − 0.696i)8-s + (0.105 − 0.994i)9-s + (0.999 + 0.0248i)10-s + (−0.944 + 0.329i)11-s + (0.980 − 0.197i)12-s + (0.437 + 0.899i)13-s + (0.901 + 0.432i)14-s + (0.827 + 0.561i)15-s + (0.481 + 0.876i)16-s + (−0.673 − 0.739i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.511 - 0.859i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.511 - 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06221802023 - 0.1094412680i\)
\(L(\frac12)\) \(\approx\) \(0.06221802023 - 0.1094412680i\)
\(L(1)\) \(\approx\) \(0.4928280346 + 0.1429149788i\)
\(L(1)\) \(\approx\) \(0.4928280346 + 0.1429149788i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.263 + 0.964i)T \)
3 \( 1 + (-0.743 + 0.668i)T \)
5 \( 1 + (-0.239 - 0.970i)T \)
7 \( 1 + (0.179 - 0.983i)T \)
11 \( 1 + (-0.944 + 0.329i)T \)
13 \( 1 + (0.437 + 0.899i)T \)
17 \( 1 + (-0.673 - 0.739i)T \)
19 \( 1 + (0.992 + 0.123i)T \)
29 \( 1 + (-0.191 - 0.981i)T \)
31 \( 1 + (-0.535 - 0.844i)T \)
37 \( 1 + (-0.709 + 0.704i)T \)
41 \( 1 + (-0.834 + 0.551i)T \)
43 \( 1 + (0.251 + 0.967i)T \)
47 \( 1 + (-0.990 - 0.136i)T \)
53 \( 1 + (0.999 - 0.0248i)T \)
59 \( 1 + (-0.986 - 0.160i)T \)
61 \( 1 + (-0.982 - 0.185i)T \)
67 \( 1 + (0.813 - 0.581i)T \)
71 \( 1 + (-0.404 - 0.914i)T \)
73 \( 1 + (-0.191 + 0.981i)T \)
79 \( 1 + (-0.993 + 0.111i)T \)
83 \( 1 + (0.995 - 0.0991i)T \)
89 \( 1 + (0.975 + 0.221i)T \)
97 \( 1 + (-0.426 + 0.904i)T \)
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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

−23.49800405286275553347348785588, −22.75220010436591677369460888188, −22.01913458838642525703369758931, −21.53621362934503530210647454480, −20.27076432108244751188202729521, −19.34099550326528226648365481281, −18.57851071370887340926518243355, −18.07652343491572850830353013075, −17.61602664566823500508402040445, −16.175619262072662726518422809442, −15.33031376602814047689486113629, −14.08004363212252935358532357971, −13.17435294537687054297409936265, −12.41722875903631265060663852719, −11.61347091979295795374322334219, −10.75366294031058378440253228902, −10.40805862576184714816501723115, −8.84038949171077507105829095057, −8.00896590787058024470728270731, −7.10706724161069216082570711294, −5.76139015974006456161991724255, −5.11736084013552477755964424226, −3.46542773119545898697713937676, −2.61820542629462828290720276977, −1.621835284261603322637564253421, 0.08756906662149420164666525313, 1.337218779978963321214144884098, 3.76551307246583870798586106780, 4.63896597304319546865916750751, 5.1109795113636807036697802038, 6.27036372070350196262726142722, 7.28507957939601204351218680310, 8.14466485308661034297998725271, 9.32231757199451639121997023644, 9.848384345077400456427947093828, 10.96663038872993540447839178435, 11.85654521301434413952582210265, 13.1949394416580203364262681263, 13.72827525535901173625394045646, 15.03070897953799438860202352284, 15.87138985992536490210601182706, 16.38038325603515028813992490943, 17.033333900089781790279350931163, 17.84415309162472850105829387066, 18.63868388576878092047134921616, 20.04484644711901227195707253747, 20.65720375217052650576374329352, 21.572816949019433525662809134944, 22.87045212556638359632705960171, 23.157294537959385620426999054640

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