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Properties

Label 1-4729-4729.3309-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.775 - 0.631i$
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  + (0.290 + 0.956i)2-s + (0.0398 − 0.999i)3-s + (−0.830 + 0.556i)4-s + (−0.998 − 0.0478i)5-s + (0.967 − 0.252i)6-s + (0.721 + 0.692i)7-s + (−0.773 − 0.633i)8-s + (−0.996 − 0.0796i)9-s + (−0.244 − 0.969i)10-s + (0.481 + 0.876i)11-s + (0.522 + 0.852i)12-s + (−0.753 − 0.657i)13-s + (−0.453 + 0.891i)14-s + (−0.0875 + 0.996i)15-s + (0.380 − 0.924i)16-s + (−0.229 + 0.973i)17-s + ⋯
L(s)  = 1  + (0.290 + 0.956i)2-s + (0.0398 − 0.999i)3-s + (−0.830 + 0.556i)4-s + (−0.998 − 0.0478i)5-s + (0.967 − 0.252i)6-s + (0.721 + 0.692i)7-s + (−0.773 − 0.633i)8-s + (−0.996 − 0.0796i)9-s + (−0.244 − 0.969i)10-s + (0.481 + 0.876i)11-s + (0.522 + 0.852i)12-s + (−0.753 − 0.657i)13-s + (−0.453 + 0.891i)14-s + (−0.0875 + 0.996i)15-s + (0.380 − 0.924i)16-s + (−0.229 + 0.973i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.775 - 0.631i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (3309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.775 - 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1633754596 + 0.4592903135i\)
\(L(\frac12)\) \(\approx\) \(-0.1633754596 + 0.4592903135i\)
\(L(1)\) \(\approx\) \(0.7454327903 + 0.3752638620i\)
\(L(1)\) \(\approx\) \(0.7454327903 + 0.3752638620i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.290 + 0.956i)T \)
3 \( 1 + (0.0398 - 0.999i)T \)
5 \( 1 + (-0.998 - 0.0478i)T \)
7 \( 1 + (0.721 + 0.692i)T \)
11 \( 1 + (0.481 + 0.876i)T \)
13 \( 1 + (-0.753 - 0.657i)T \)
17 \( 1 + (-0.229 + 0.973i)T \)
19 \( 1 + (0.576 + 0.817i)T \)
23 \( 1 + (-0.00797 + 0.999i)T \)
29 \( 1 + (-0.290 + 0.956i)T \)
31 \( 1 + (-0.987 - 0.158i)T \)
37 \( 1 + (-0.305 - 0.952i)T \)
41 \( 1 + (-0.915 - 0.402i)T \)
43 \( 1 + (0.732 + 0.681i)T \)
47 \( 1 + (0.709 - 0.704i)T \)
53 \( 1 + (-0.549 + 0.835i)T \)
59 \( 1 + (0.351 + 0.936i)T \)
61 \( 1 + (-0.166 - 0.986i)T \)
67 \( 1 + (-0.971 - 0.236i)T \)
71 \( 1 + (0.997 - 0.0637i)T \)
73 \( 1 + (-0.366 - 0.930i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.275 + 0.961i)T \)
89 \( 1 + (0.424 - 0.905i)T \)
97 \( 1 + (0.721 + 0.692i)T \)
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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

−17.75368174396056123435215420079, −17.00289575817112944155720476010, −16.453502636264599622880945206285, −15.64166245382378174771880993513, −14.95295869115532631952226300750, −14.18444314755446364583862094662, −14.03488342723202409645586608706, −13.04154791891953951582622297039, −11.88955780779220507793087329159, −11.56899146950717141660174293422, −11.15244639179688965283087939532, −10.45666653715156768517533070464, −9.7038484313603245113738358954, −8.941587542579238760846197336099, −8.48604860209030205110973227249, −7.54003804249912759897293644652, −6.68433011363113170233479151066, −5.51925435472107156185305620110, −4.69267804913621552840159059436, −4.47341527249604854056771952928, −3.66905350031666746764486827821, −3.04752974632276829422173203691, −2.23254300855799586088663190876, −0.928637650890091145241518985816, −0.14649356733895585843140726260, 1.26519976465588644002261215410, 2.086096554440370438915999916347, 3.22596209324692825959423659723, 3.84925585016485142605440486296, 4.83559887436897161227485436998, 5.46430059625480865233073403454, 6.0858275731500177481818207403, 7.17673108564602825989933497909, 7.48137743678663175597682041745, 7.96927431002049715936391081354, 8.78854783148983370482247348669, 9.25864891624217556434693167970, 10.50821178964324018237969253858, 11.493445824942126215870028491267, 12.149244468226021663659563500533, 12.49433497561863628687344028899, 13.037414141975676131283555924595, 14.21665610267775246634913044557, 14.54536583350742589413395948145, 15.18347901061165611586941721242, 15.63325225731991313145349464397, 16.71261179894598163983908318702, 17.2119318150449171055394357252, 17.90509867171164082029446621769, 18.34361434486185838052116927821

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