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Properties

Label 1-4729-4729.1040-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.984 + 0.177i$
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  + (−0.856 − 0.516i)2-s + (−0.821 − 0.569i)3-s + (0.467 + 0.884i)4-s + (0.739 − 0.673i)5-s + (0.410 + 0.912i)6-s + (−0.735 − 0.677i)7-s + (0.0557 − 0.998i)8-s + (0.351 + 0.936i)9-s + (−0.980 + 0.195i)10-s + (0.0955 + 0.995i)11-s + (0.119 − 0.992i)12-s + (−0.818 + 0.573i)13-s + (0.280 + 0.959i)14-s + (−0.991 + 0.132i)15-s + (−0.563 + 0.826i)16-s + (−0.711 + 0.702i)17-s + ⋯
L(s)  = 1  + (−0.856 − 0.516i)2-s + (−0.821 − 0.569i)3-s + (0.467 + 0.884i)4-s + (0.739 − 0.673i)5-s + (0.410 + 0.912i)6-s + (−0.735 − 0.677i)7-s + (0.0557 − 0.998i)8-s + (0.351 + 0.936i)9-s + (−0.980 + 0.195i)10-s + (0.0955 + 0.995i)11-s + (0.119 − 0.992i)12-s + (−0.818 + 0.573i)13-s + (0.280 + 0.959i)14-s + (−0.991 + 0.132i)15-s + (−0.563 + 0.826i)16-s + (−0.711 + 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.984 + 0.177i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1040, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.984 + 0.177i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02583205090 - 0.2892287521i\)
\(L(\frac12)\) \(\approx\) \(0.02583205090 - 0.2892287521i\)
\(L(1)\) \(\approx\) \(0.4507886744 - 0.2216957852i\)
\(L(1)\) \(\approx\) \(0.4507886744 - 0.2216957852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.856 - 0.516i)T \)
3 \( 1 + (-0.821 - 0.569i)T \)
5 \( 1 + (0.739 - 0.673i)T \)
7 \( 1 + (-0.735 - 0.677i)T \)
11 \( 1 + (0.0955 + 0.995i)T \)
13 \( 1 + (-0.818 + 0.573i)T \)
17 \( 1 + (-0.711 + 0.702i)T \)
19 \( 1 + (0.993 - 0.116i)T \)
23 \( 1 + (-0.681 + 0.732i)T \)
29 \( 1 + (-0.516 - 0.856i)T \)
31 \( 1 + (-0.657 - 0.753i)T \)
37 \( 1 + (-0.995 - 0.0902i)T \)
41 \( 1 + (0.956 - 0.290i)T \)
43 \( 1 + (-0.288 + 0.957i)T \)
47 \( 1 + (-0.148 - 0.988i)T \)
53 \( 1 + (-0.257 + 0.966i)T \)
59 \( 1 + (0.976 - 0.216i)T \)
61 \( 1 + (0.881 + 0.472i)T \)
67 \( 1 + (0.851 - 0.525i)T \)
71 \( 1 + (0.724 - 0.689i)T \)
73 \( 1 + (0.929 + 0.368i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (0.998 + 0.0531i)T \)
89 \( 1 + (-0.990 + 0.140i)T \)
97 \( 1 + (0.735 + 0.677i)T \)
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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

−18.25736423420354279369156972787, −17.92281092989305753500514712605, −17.18888870022370638567386757423, −16.489500178187561245871846499990, −15.90001478573082904055117446738, −15.56936735824483571325132538936, −14.48027138956771809825673637807, −14.26250431736570682895317341559, −13.110457897452068027969416442887, −12.28899869500274321838667673914, −11.44674188763459189679190870940, −10.93329322087779266720940276263, −10.19316440848508406202645403600, −9.70935600603542532900044957616, −9.15046914490346854631741363077, −8.45252660341030415607459772116, −7.2217350832251919848569377787, −6.765211094662317030147389083193, −6.103905895561003870293886218605, −5.36075156977252708013596241828, −5.17001654882107427455965645191, −3.585579756391269125856584853482, −2.86445855324942904473092464795, −2.05738582359244190674108117779, −0.82737448200270990337277827822, 0.15531747642212336426615142951, 1.1289427588908295163273449462, 1.96436428390424352908957508329, 2.322620824567965606529708153096, 3.77176518234459475648628017155, 4.40815057989905634409618468958, 5.33948260700799031921115158687, 6.21803599259391051705767863309, 6.88247837274684018359405072522, 7.4337953749823580037642255489, 8.10843676655733536794859290297, 9.34000569198397735528086207388, 9.606992885121307160080741789250, 10.20669930974432350926304779812, 10.9752597923078982298829121120, 11.828709718090062177061859725006, 12.26778487787408696451703582921, 13.05859170925295835452694872267, 13.30622120589136822670906299014, 14.22568180456045778892927409364, 15.513820572999019017036865639769, 16.13395190447505457086119098125, 16.82678334640341466419344973268, 17.25230362950786164676268266737, 17.66529438395647465252917084469

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