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Properties

Label 1-4729-4729.1065-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.878 + 0.476i$
Analytic cond. $21.9613$
Root an. cond. $21.9613$
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 + (−0.495 − 0.868i)3-s + (−0.182 − 0.983i)4-s + (0.812 − 0.582i)5-s + (−0.984 − 0.174i)6-s + (0.864 − 0.502i)7-s + (−0.872 − 0.488i)8-s + (−0.509 + 0.860i)9-s + (0.0717 − 0.997i)10-s + (−0.969 − 0.244i)11-s + (−0.763 + 0.645i)12-s + (0.995 + 0.0955i)13-s + (0.166 − 0.986i)14-s + (−0.908 − 0.417i)15-s + (−0.933 + 0.358i)16-s + (−0.990 − 0.135i)17-s + ⋯
L(s)  = 1  + (0.639 − 0.768i)2-s + (−0.495 − 0.868i)3-s + (−0.182 − 0.983i)4-s + (0.812 − 0.582i)5-s + (−0.984 − 0.174i)6-s + (0.864 − 0.502i)7-s + (−0.872 − 0.488i)8-s + (−0.509 + 0.860i)9-s + (0.0717 − 0.997i)10-s + (−0.969 − 0.244i)11-s + (−0.763 + 0.645i)12-s + (0.995 + 0.0955i)13-s + (0.166 − 0.986i)14-s + (−0.908 − 0.417i)15-s + (−0.933 + 0.358i)16-s + (−0.990 − 0.135i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.878 + 0.476i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1065, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.878 + 0.476i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6530130751 - 2.572521367i\)
\(L(\frac12)\) \(\approx\) \(-0.6530130751 - 2.572521367i\)
\(L(1)\) \(\approx\) \(0.7855963368 - 1.316728711i\)
\(L(1)\) \(\approx\) \(0.7855963368 - 1.316728711i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.639 - 0.768i)T \)
3 \( 1 + (-0.495 - 0.868i)T \)
5 \( 1 + (0.812 - 0.582i)T \)
7 \( 1 + (0.864 - 0.502i)T \)
11 \( 1 + (-0.969 - 0.244i)T \)
13 \( 1 + (0.995 + 0.0955i)T \)
17 \( 1 + (-0.990 - 0.135i)T \)
19 \( 1 + (0.991 + 0.127i)T \)
23 \( 1 + (0.994 - 0.103i)T \)
29 \( 1 + (-0.768 + 0.639i)T \)
31 \( 1 + (0.876 - 0.481i)T \)
37 \( 1 + (0.783 - 0.620i)T \)
41 \( 1 + (0.778 - 0.627i)T \)
43 \( 1 + (-0.313 - 0.949i)T \)
47 \( 1 + (0.669 - 0.742i)T \)
53 \( 1 + (0.283 - 0.959i)T \)
59 \( 1 + (0.998 - 0.0478i)T \)
61 \( 1 + (0.569 - 0.821i)T \)
67 \( 1 + (0.999 + 0.0318i)T \)
71 \( 1 + (-0.675 - 0.737i)T \)
73 \( 1 + (-0.987 - 0.158i)T \)
79 \( 1 + iT \)
83 \( 1 + (0.467 + 0.884i)T \)
89 \( 1 + (-0.835 - 0.549i)T \)
97 \( 1 + (-0.864 + 0.502i)T \)
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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

−18.08711810153571543750311183303, −17.77195526139036686968942664186, −17.296714502947459748202582439791, −16.33718958021976862092828660677, −15.76733094062670652003369123763, −15.1383391537891823364435631891, −14.81658989681896871468386227606, −13.91788681966612433435155507465, −13.33818917767252635552063386092, −12.711038316481339020627167490175, −11.47230829750458653968995239444, −11.338203751838452045090496200824, −10.526920065570169926184267182220, −9.61908717456352941212501308049, −8.96871059889012014312364159218, −8.271386624063685701703462743507, −7.41592625718975757094609493979, −6.57108581223499069980015371367, −5.83545047899619299666289743148, −5.46868792181422219039554717664, −4.71237424898021890820991026466, −4.12747715647850245895388346566, −2.892210555437429507049401153064, −2.68248233779855651916905385343, −1.26119845806149076866366089529, 0.67054885242899731483449441716, 1.14108052668372626327086942916, 2.0642669978782130757953377033, 2.50390510933623721139479010676, 3.69594904767029092446741404446, 4.617145945773067907448194633604, 5.38953036645017398653675965775, 5.55322594273967436165837536754, 6.57057095179326026108339064555, 7.25593401015090270353472194308, 8.29655343506088968336683588475, 8.836682659836018853904531927987, 9.788435454077723330238502192302, 10.707959969740359747370939943198, 11.03634649256269098101943302932, 11.64436155213506938041212145983, 12.53204764389976998378687840888, 13.181432229165864513530903527229, 13.5363120328074612963885823091, 13.96278084818320744114856817645, 14.83860841783178132863673043782, 15.82711017396430003763163699578, 16.46148205409956733537085169502, 17.39305209927298387753689508861, 17.91608322966163739302969020850

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