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Properties

Label 1-4729-4729.1083-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.310 + 0.950i$
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  + (−0.103 + 0.994i)2-s + (0.424 + 0.905i)3-s + (−0.978 − 0.205i)4-s + (−0.867 − 0.497i)5-s + (−0.944 + 0.328i)6-s + (0.462 − 0.886i)7-s + (0.305 − 0.952i)8-s + (−0.639 + 0.768i)9-s + (0.584 − 0.811i)10-s + (−0.726 + 0.687i)11-s + (−0.229 − 0.973i)12-s + (0.845 − 0.534i)13-s + (0.833 + 0.551i)14-s + (0.0823 − 0.996i)15-s + (0.915 + 0.402i)16-s + (−0.0743 + 0.997i)17-s + ⋯
L(s)  = 1  + (−0.103 + 0.994i)2-s + (0.424 + 0.905i)3-s + (−0.978 − 0.205i)4-s + (−0.867 − 0.497i)5-s + (−0.944 + 0.328i)6-s + (0.462 − 0.886i)7-s + (0.305 − 0.952i)8-s + (−0.639 + 0.768i)9-s + (0.584 − 0.811i)10-s + (−0.726 + 0.687i)11-s + (−0.229 − 0.973i)12-s + (0.845 − 0.534i)13-s + (0.833 + 0.551i)14-s + (0.0823 − 0.996i)15-s + (0.915 + 0.402i)16-s + (−0.0743 + 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.310 + 0.950i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1083, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.310 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7890597819 + 1.088267383i\)
\(L(\frac12)\) \(\approx\) \(0.7890597819 + 1.088267383i\)
\(L(1)\) \(\approx\) \(0.7540882458 + 0.5519181964i\)
\(L(1)\) \(\approx\) \(0.7540882458 + 0.5519181964i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.103 + 0.994i)T \)
3 \( 1 + (0.424 + 0.905i)T \)
5 \( 1 + (-0.867 - 0.497i)T \)
7 \( 1 + (0.462 - 0.886i)T \)
11 \( 1 + (-0.726 + 0.687i)T \)
13 \( 1 + (0.845 - 0.534i)T \)
17 \( 1 + (-0.0743 + 0.997i)T \)
19 \( 1 + (0.998 - 0.0531i)T \)
23 \( 1 + (0.996 - 0.0875i)T \)
29 \( 1 + (-0.994 + 0.103i)T \)
31 \( 1 + (-0.983 + 0.182i)T \)
37 \( 1 + (0.970 - 0.242i)T \)
41 \( 1 + (-0.988 - 0.150i)T \)
43 \( 1 + (0.132 - 0.991i)T \)
47 \( 1 + (0.216 - 0.976i)T \)
53 \( 1 + (-0.393 + 0.919i)T \)
59 \( 1 + (-0.239 - 0.970i)T \)
61 \( 1 + (0.700 + 0.713i)T \)
67 \( 1 + (-0.859 + 0.511i)T \)
71 \( 1 + (0.940 - 0.338i)T \)
73 \( 1 + (-0.897 - 0.441i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.827 + 0.560i)T \)
89 \( 1 + (-0.804 + 0.593i)T \)
97 \( 1 + (-0.462 + 0.886i)T \)
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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

−18.34432835896940945009325557480, −17.851243032954343965758309463879, −16.65328396707912186149561095358, −15.94177227422760126267052424460, −15.04822196866982872448991175893, −14.467684829912213560141628514044, −13.77286540506169908821410636904, −13.15704474769709813494881489638, −12.5360794654737871763002704459, −11.60655609920565422410972654311, −11.400335869940058194758680840797, −10.95218580186505687961047847409, −9.61830156364793882836247461124, −9.04040664843189548698828300543, −8.40936376735247862089103684290, −7.78230504786703798935377266358, −7.23826620658891887039942097435, −6.114789754229798962899166263994, −5.36912339330221363594474131905, −4.52132429226395353781253078697, −3.32261042117048894946505409713, −3.13910617033209990093421379131, −2.3206968076120324408747002338, −1.46540781506974977673117225454, −0.57532435966457777544312688348, 0.67393817455826385990859114111, 1.719856105444317446762440282879, 3.282207909247342741626997582318, 3.79480882767114962074714551254, 4.380739266153925699212093475404, 5.20618438990487343897027240086, 5.50926258453290375777134370296, 6.87109057348225584680441863233, 7.63531390490022428064829996541, 7.92668064196922738872429758139, 8.71683702114537413726782811086, 9.24910660132083083007613770570, 10.20413827650079781177977487775, 10.70893499654277235613749504311, 11.353076303882505778093277731549, 12.59369593822429325654490299797, 13.19905427448047310109828854794, 13.75214935047713514320749426717, 14.69036881422464082887748124491, 15.203849790730235244629977775792, 15.51814678355740402039559833399, 16.44218080170735976138579651863, 16.6912786882725192428841890597, 17.47015778137892588945950797520, 18.223931511647171006396279767074

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