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Properties

Label 1-4729-4729.64-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.958 + 0.286i$
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  + (−0.481 + 0.876i)2-s + (−0.793 − 0.608i)3-s + (−0.536 − 0.843i)4-s + (−0.453 + 0.891i)5-s + (0.915 − 0.402i)6-s + (0.321 + 0.947i)7-s + (0.997 − 0.0637i)8-s + (0.260 + 0.965i)9-s + (−0.563 − 0.826i)10-s + (0.848 + 0.529i)11-s + (−0.0875 + 0.996i)12-s + (0.698 − 0.715i)13-s + (−0.984 − 0.174i)14-s + (0.901 − 0.431i)15-s + (−0.424 + 0.905i)16-s + (−0.821 + 0.569i)17-s + ⋯
L(s)  = 1  + (−0.481 + 0.876i)2-s + (−0.793 − 0.608i)3-s + (−0.536 − 0.843i)4-s + (−0.453 + 0.891i)5-s + (0.915 − 0.402i)6-s + (0.321 + 0.947i)7-s + (0.997 − 0.0637i)8-s + (0.260 + 0.965i)9-s + (−0.563 − 0.826i)10-s + (0.848 + 0.529i)11-s + (−0.0875 + 0.996i)12-s + (0.698 − 0.715i)13-s + (−0.984 − 0.174i)14-s + (0.901 − 0.431i)15-s + (−0.424 + 0.905i)16-s + (−0.821 + 0.569i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.958 + 0.286i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.958 + 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1230526084 + 0.8419243441i\)
\(L(\frac12)\) \(\approx\) \(0.1230526084 + 0.8419243441i\)
\(L(1)\) \(\approx\) \(0.5325767666 + 0.3690871126i\)
\(L(1)\) \(\approx\) \(0.5325767666 + 0.3690871126i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.481 + 0.876i)T \)
3 \( 1 + (-0.793 - 0.608i)T \)
5 \( 1 + (-0.453 + 0.891i)T \)
7 \( 1 + (0.321 + 0.947i)T \)
11 \( 1 + (0.848 + 0.529i)T \)
13 \( 1 + (0.698 - 0.715i)T \)
17 \( 1 + (-0.821 + 0.569i)T \)
19 \( 1 + (-0.999 + 0.0159i)T \)
23 \( 1 + (0.182 + 0.983i)T \)
29 \( 1 + (0.481 + 0.876i)T \)
31 \( 1 + (0.864 - 0.502i)T \)
37 \( 1 + (0.763 - 0.645i)T \)
41 \( 1 + (0.993 - 0.111i)T \)
43 \( 1 + (-0.0398 + 0.999i)T \)
47 \( 1 + (0.639 - 0.768i)T \)
53 \( 1 + (0.732 + 0.681i)T \)
59 \( 1 + (-0.921 - 0.388i)T \)
61 \( 1 + (-0.651 + 0.758i)T \)
67 \( 1 + (-0.709 - 0.704i)T \)
71 \( 1 + (0.103 + 0.994i)T \)
73 \( 1 + (0.721 + 0.692i)T \)
79 \( 1 - T \)
83 \( 1 + (0.135 + 0.990i)T \)
89 \( 1 + (0.614 + 0.788i)T \)
97 \( 1 + (0.321 + 0.947i)T \)
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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

−17.64110064633077501469223357939, −17.17774657158517220864177205893, −16.66100299330103867006004597385, −16.219502247606535269648142881076, −15.45511492652439543245680820509, −14.35379796671321065210461381245, −13.62236374835151192758944801149, −13.02199140825851666732901164274, −12.09625286719205631161178971140, −11.70777153937555785844285941226, −11.03028004822625749883407319454, −10.61971905168763944335608354850, −9.75007851828254581982591571050, −8.92234116534802234077227867256, −8.68026102385899088275947977914, −7.72765429292371625283753393395, −6.73771094968316867467722785384, −6.145535594648055525086825835549, −4.78394874109936238225677797948, −4.32610020365886543471419245188, −4.10618823378921253580829691344, −3.10631867692931169951646038039, −1.82498756052907394915191649533, −0.93959678067789395230084406374, −0.45763395347400283219682490216, 0.96713965462767594932843869964, 1.82390515191235499382716048964, 2.63531771839887163505317375307, 4.01532111832122357350157264999, 4.61120874000540968625417147830, 5.68105914298420877452045682983, 6.126856148760514641943043740993, 6.65774268061543967620070705020, 7.38956793311381037301369463138, 8.061763107400827211017195121815, 8.6721944394172024736598930188, 9.5032244521788741018000224149, 10.49932328459282960255132455833, 10.9570717774151980088269690194, 11.55891725134553028359408153042, 12.41579921720415019008918569037, 13.07138176640446427141251044897, 13.92498917167631069622448050916, 14.66805822786825393905924441156, 15.38084470979029893813780473145, 15.55611575355867566346694603432, 16.56816060880587813546560606397, 17.29677610284681182801716130010, 17.94688988433048048917527499814, 18.10424243246054259442842160059

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