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Properties

Label 1-4729-4729.3615-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.766 + 0.642i$
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  + (−0.999 − 0.0159i)2-s + (0.290 − 0.956i)3-s + (0.999 + 0.0318i)4-s + (−0.885 + 0.465i)5-s + (−0.305 + 0.952i)6-s + (0.114 − 0.993i)7-s + (−0.998 − 0.0478i)8-s + (−0.830 − 0.556i)9-s + (0.892 − 0.450i)10-s + (−0.933 − 0.358i)11-s + (0.321 − 0.947i)12-s + (0.997 + 0.0743i)13-s + (−0.129 + 0.991i)14-s + (0.187 + 0.982i)15-s + (0.997 + 0.0637i)16-s + (−0.997 + 0.0690i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0159i)2-s + (0.290 − 0.956i)3-s + (0.999 + 0.0318i)4-s + (−0.885 + 0.465i)5-s + (−0.305 + 0.952i)6-s + (0.114 − 0.993i)7-s + (−0.998 − 0.0478i)8-s + (−0.830 − 0.556i)9-s + (0.892 − 0.450i)10-s + (−0.933 − 0.358i)11-s + (0.321 − 0.947i)12-s + (0.997 + 0.0743i)13-s + (−0.129 + 0.991i)14-s + (0.187 + 0.982i)15-s + (0.997 + 0.0637i)16-s + (−0.997 + 0.0690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (3615, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2623847521 + 0.09545266010i\)
\(L(\frac12)\) \(\approx\) \(0.2623847521 + 0.09545266010i\)
\(L(1)\) \(\approx\) \(0.5021346857 - 0.2113289113i\)
\(L(1)\) \(\approx\) \(0.5021346857 - 0.2113289113i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.999 - 0.0159i)T \)
3 \( 1 + (0.290 - 0.956i)T \)
5 \( 1 + (-0.885 + 0.465i)T \)
7 \( 1 + (0.114 - 0.993i)T \)
11 \( 1 + (-0.933 - 0.358i)T \)
13 \( 1 + (0.997 + 0.0743i)T \)
17 \( 1 + (-0.997 + 0.0690i)T \)
19 \( 1 + (0.270 - 0.962i)T \)
23 \( 1 + (0.967 - 0.252i)T \)
29 \( 1 + (-0.999 + 0.0159i)T \)
31 \( 1 + (0.380 + 0.924i)T \)
37 \( 1 + (-0.00265 + 0.999i)T \)
41 \( 1 + (0.763 - 0.645i)T \)
43 \( 1 + (-0.994 - 0.100i)T \)
47 \( 1 + (-0.606 - 0.795i)T \)
53 \( 1 + (-0.679 + 0.733i)T \)
59 \( 1 + (0.531 + 0.846i)T \)
61 \( 1 + (0.390 + 0.920i)T \)
67 \( 1 + (0.749 + 0.661i)T \)
71 \( 1 + (-0.545 - 0.838i)T \)
73 \( 1 + (-0.890 - 0.455i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.851 - 0.525i)T \)
89 \( 1 + (-0.913 + 0.407i)T \)
97 \( 1 + (0.114 - 0.993i)T \)
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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

−18.1313650896799476639447149323, −17.42139781296644726738345817823, −16.556263712504841407429693802297, −15.911119916319573777571338737547, −15.720535039798190730339952360051, −15.042114870923265062364034426520, −14.53904206085154987578697119462, −13.125696807135242443212965135869, −12.688730597886280341226825290203, −11.49049838153862586577958009127, −11.330859718613998243179512938791, −10.66530389094168039479040066679, −9.617119197744408594140999887374, −9.29649140193786642450323053125, −8.39844366569142826058511423555, −8.1821203269416646105207231036, −7.429912076512524251694933323420, −6.29150916613612402224287150637, −5.51934218871363199665786981381, −4.89133467414888629842812746796, −3.88232240164117059992085732663, −3.18110644124110426782388287669, −2.39950911099517648456018284851, −1.54879396722971218495643480611, −0.13629960052591814291782739582, 0.75189188228955859925964222553, 1.479464007802801585113427322188, 2.60137566422745451226469628878, 3.10999632464995039326104772585, 3.898617083515207803344600178081, 5.06895832963080593791128343020, 6.2089980245730893179093127895, 6.91174266617274412444364933177, 7.19551642119282060583567637110, 7.96460882796437726616484499633, 8.54313240641978995065599677889, 9.034755273592878976100191094402, 10.27044108349293159690517750249, 10.88854770950018562180743132882, 11.28047487725455090802259897736, 11.88922498493568410062683240542, 13.07743807296905844762546187459, 13.27644843468236291515110474608, 14.230687368661224197659777846652, 15.099712846271928584336483019188, 15.58611486716944132462039960724, 16.32943295218761775065066571974, 17.02185224716763540717912699782, 17.894461380731784015846166998189, 18.15774382856339520876307187502

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