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Properties

Label 1-4729-4729.1043-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.996 - 0.0853i$
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  + (−0.522 + 0.852i)2-s + (−0.999 + 0.0318i)3-s + (−0.453 − 0.891i)4-s + (0.0663 − 0.997i)5-s + (0.495 − 0.868i)6-s + (−0.486 − 0.873i)7-s + (0.996 + 0.0796i)8-s + (0.997 − 0.0637i)9-s + (0.815 + 0.578i)10-s + (0.996 + 0.0875i)11-s + (0.481 + 0.876i)12-s + (−0.890 − 0.455i)13-s + (0.999 + 0.0425i)14-s + (−0.0345 + 0.999i)15-s + (−0.589 + 0.808i)16-s + (−0.231 + 0.972i)17-s + ⋯
L(s)  = 1  + (−0.522 + 0.852i)2-s + (−0.999 + 0.0318i)3-s + (−0.453 − 0.891i)4-s + (0.0663 − 0.997i)5-s + (0.495 − 0.868i)6-s + (−0.486 − 0.873i)7-s + (0.996 + 0.0796i)8-s + (0.997 − 0.0637i)9-s + (0.815 + 0.578i)10-s + (0.996 + 0.0875i)11-s + (0.481 + 0.876i)12-s + (−0.890 − 0.455i)13-s + (0.999 + 0.0425i)14-s + (−0.0345 + 0.999i)15-s + (−0.589 + 0.808i)16-s + (−0.231 + 0.972i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.996 - 0.0853i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.996 - 0.0853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6205789177 + 0.02651864195i\)
\(L(\frac12)\) \(\approx\) \(0.6205789177 + 0.02651864195i\)
\(L(1)\) \(\approx\) \(0.5390848846 + 0.03028277471i\)
\(L(1)\) \(\approx\) \(0.5390848846 + 0.03028277471i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.522 + 0.852i)T \)
3 \( 1 + (-0.999 + 0.0318i)T \)
5 \( 1 + (0.0663 - 0.997i)T \)
7 \( 1 + (-0.486 - 0.873i)T \)
11 \( 1 + (0.996 + 0.0875i)T \)
13 \( 1 + (-0.890 - 0.455i)T \)
17 \( 1 + (-0.231 + 0.972i)T \)
19 \( 1 + (-0.960 - 0.278i)T \)
23 \( 1 + (0.582 - 0.812i)T \)
29 \( 1 + (-0.852 + 0.522i)T \)
31 \( 1 + (-0.127 - 0.991i)T \)
37 \( 1 + (-0.346 + 0.938i)T \)
41 \( 1 + (-0.798 + 0.601i)T \)
43 \( 1 + (0.762 - 0.647i)T \)
47 \( 1 + (0.211 + 0.977i)T \)
53 \( 1 + (-0.363 + 0.931i)T \)
59 \( 1 + (0.724 - 0.689i)T \)
61 \( 1 + (0.999 + 0.0292i)T \)
67 \( 1 + (-0.326 + 0.945i)T \)
71 \( 1 + (0.966 + 0.257i)T \)
73 \( 1 + (0.995 + 0.0902i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (-0.295 + 0.955i)T \)
89 \( 1 + (0.641 - 0.767i)T \)
97 \( 1 + (0.486 + 0.873i)T \)
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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

−18.17877189471568337680003965658, −17.63926909662599953218016693378, −17.03574931146885552419615083400, −16.40773901851153814274744483855, −15.627835515825796086832355983119, −14.84543067466209896294891651058, −14.071695362169033896176507745256, −13.20474783511355927389807979285, −12.49172400301320663621554531095, −11.803909062706346243128523387439, −11.54192765127378499778997972226, −10.78327279373193844733659004722, −10.10008925505492231562983613797, −9.383236285762170598424005099900, −9.04148064951692421688696308375, −7.79062746237901307002750691481, −6.91592573927889607312770423221, −6.70645272405597343365225300140, −5.63334109407306157193625935613, −4.90280871989277148182898246280, −3.89486340364380987292688188859, −3.331480316783098570872524312032, −2.20422319307004195966055323744, −1.89248681546324731024549139395, −0.486699956743632613939432574207, 0.519488928378287677508233705, 1.17161683757572445170088724308, 2.050507737783961225045730929549, 3.85565326568275883340482606295, 4.34871140473569829575265200906, 4.98635559556094842465870042330, 5.77085641706673429765343120926, 6.50416121207172234886521661292, 6.92298187216868776899733347628, 7.779465173370124459723618691855, 8.53318618259188128783218445079, 9.38656208514919664205997563976, 9.83324700305880895185223956377, 10.60746215592124508054179509465, 11.15461663617691921357790190740, 12.233977271057235925668074063655, 12.87744895851934126017800224897, 13.24083119931618782140756178898, 14.2982221725263582839432673234, 15.04255703558516905990348282294, 15.61005668398462870566821035930, 16.5673386064455656816706900663, 16.87615219750903053511156701097, 17.2251620383185539079070236833, 17.64691139715581094831043661795

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