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Properties

Label 1-4693-4693.71-r0-0-0
Degree $1$
Conductor $4693$
Sign $-0.562 - 0.826i$
Analytic cond. $21.7942$
Root an. cond. $21.7942$
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  + (−0.599 − 0.800i)2-s + (0.155 − 0.987i)3-s + (−0.280 + 0.959i)4-s + (−0.539 + 0.842i)5-s + (−0.883 + 0.467i)6-s + (0.110 + 0.993i)7-s + (0.936 − 0.350i)8-s + (−0.951 − 0.307i)9-s + (0.997 − 0.0734i)10-s + (−0.990 + 0.137i)11-s + (0.904 + 0.426i)12-s + (0.729 − 0.684i)14-s + (0.748 + 0.663i)15-s + (−0.842 − 0.539i)16-s + (0.971 + 0.236i)17-s + (0.324 + 0.945i)18-s + ⋯
L(s)  = 1  + (−0.599 − 0.800i)2-s + (0.155 − 0.987i)3-s + (−0.280 + 0.959i)4-s + (−0.539 + 0.842i)5-s + (−0.883 + 0.467i)6-s + (0.110 + 0.993i)7-s + (0.936 − 0.350i)8-s + (−0.951 − 0.307i)9-s + (0.997 − 0.0734i)10-s + (−0.990 + 0.137i)11-s + (0.904 + 0.426i)12-s + (0.729 − 0.684i)14-s + (0.748 + 0.663i)15-s + (−0.842 − 0.539i)16-s + (0.971 + 0.236i)17-s + (0.324 + 0.945i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4693\)    =    \(13 \cdot 19^{2}\)
Sign: $-0.562 - 0.826i$
Analytic conductor: \(21.7942\)
Root analytic conductor: \(21.7942\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4693} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4693,\ (0:\ ),\ -0.562 - 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2907143517 - 0.5497860571i\)
\(L(\frac12)\) \(\approx\) \(0.2907143517 - 0.5497860571i\)
\(L(1)\) \(\approx\) \(0.5864082044 - 0.2560284105i\)
\(L(1)\) \(\approx\) \(0.5864082044 - 0.2560284105i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.599 - 0.800i)T \)
3 \( 1 + (0.155 - 0.987i)T \)
5 \( 1 + (-0.539 + 0.842i)T \)
7 \( 1 + (0.110 + 0.993i)T \)
11 \( 1 + (-0.990 + 0.137i)T \)
17 \( 1 + (0.971 + 0.236i)T \)
23 \( 1 + (0.777 - 0.628i)T \)
29 \( 1 + (-0.832 - 0.554i)T \)
31 \( 1 + (-0.0550 - 0.998i)T \)
37 \( 1 + (-0.376 + 0.926i)T \)
41 \( 1 + (0.783 - 0.621i)T \)
43 \( 1 + (0.562 - 0.826i)T \)
47 \( 1 + (-0.0367 + 0.999i)T \)
53 \( 1 + (0.467 + 0.883i)T \)
59 \( 1 + (-0.783 + 0.621i)T \)
61 \( 1 + (-0.861 - 0.507i)T \)
67 \( 1 + (-0.942 - 0.333i)T \)
71 \( 1 + (-0.492 + 0.870i)T \)
73 \( 1 + (0.723 + 0.690i)T \)
79 \( 1 + (0.562 - 0.826i)T \)
83 \( 1 + (-0.218 - 0.975i)T \)
89 \( 1 + (-0.959 - 0.280i)T \)
97 \( 1 + (0.942 - 0.333i)T \)
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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

−18.20017829535322611130185298761, −17.53012165455646980460638355692, −16.77630977627908327366571163925, −16.34809548986769849960424531679, −16.02411467013138093821669232375, −15.16471071236664582737046305806, −14.67064202690868641472885656863, −13.84145973132260615895416351336, −13.29195912060479958598730428467, −12.38062523379992407444522625966, −11.21690578309831683302098946873, −10.82713541107629404238330451037, −10.13002191561275143886054460199, −9.3982049931416749006140340709, −8.90825664576362977700487445474, −7.92743837704683054152871875114, −7.7543022018364317058451267061, −6.88170803687458537022653095542, −5.595079779942991154893001733346, −5.24696531914424902237953969965, −4.59053674161343205918308547058, −3.810258052984103776041282610228, −3.03927590219019502418513432620, −1.61817790202600747213181459482, −0.71887090197786430255951756735, 0.29239304194316139324449063384, 1.44307508413309962817199447048, 2.373176353570108012294866892, 2.73753998924016982906008841803, 3.39677563644837812788677935541, 4.44405269455937372950248143981, 5.561166845411353648750077659282, 6.22629156963889281799601813703, 7.33434680552383922407732749701, 7.62875962875325386591347125657, 8.28537350591825700421089124322, 8.98045769453553147274068702175, 9.772527123250652995625182837593, 10.70811359934398375258856720589, 11.12221203652544843060703148252, 12.02135312108733106414941370011, 12.2795841947337854152673275355, 13.03037571848656401867210815113, 13.716030899592787516194153030717, 14.58962367135389087525417147170, 15.1991618399785934662903361626, 15.93037624588443866410358710171, 16.9441406985154866731483012496, 17.493835073719518983879884882204, 18.43009525614048956021362235920

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