Properties

Label 1-4729-4729.46-r1-0-0
Degree $1$
Conductor $4729$
Sign $-0.982 + 0.185i$
Analytic cond. $508.201$
Root an. cond. $508.201$
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  + (−0.891 + 0.453i)2-s + (−0.0637 − 0.997i)3-s + (0.589 − 0.808i)4-s + (0.924 − 0.380i)5-s + (0.509 + 0.860i)6-s + (−0.0318 + 0.999i)7-s + (−0.158 + 0.987i)8-s + (−0.991 + 0.127i)9-s + (−0.651 + 0.758i)10-s + (−0.819 + 0.572i)11-s + (−0.843 − 0.536i)12-s + (−0.410 + 0.912i)13-s + (−0.424 − 0.905i)14-s + (−0.439 − 0.898i)15-s + (−0.305 − 0.952i)16-s + (0.666 − 0.745i)17-s + ⋯
L(s)  = 1  + (−0.891 + 0.453i)2-s + (−0.0637 − 0.997i)3-s + (0.589 − 0.808i)4-s + (0.924 − 0.380i)5-s + (0.509 + 0.860i)6-s + (−0.0318 + 0.999i)7-s + (−0.158 + 0.987i)8-s + (−0.991 + 0.127i)9-s + (−0.651 + 0.758i)10-s + (−0.819 + 0.572i)11-s + (−0.843 − 0.536i)12-s + (−0.410 + 0.912i)13-s + (−0.424 − 0.905i)14-s + (−0.439 − 0.898i)15-s + (−0.305 − 0.952i)16-s + (0.666 − 0.745i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.982 + 0.185i$
Analytic conductor: \(508.201\)
Root analytic conductor: \(508.201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (1:\ ),\ -0.982 + 0.185i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04368763088 + 0.4657489425i\)
\(L(\frac12)\) \(\approx\) \(0.04368763088 + 0.4657489425i\)
\(L(1)\) \(\approx\) \(0.6879606416 + 0.04785929291i\)
\(L(1)\) \(\approx\) \(0.6879606416 + 0.04785929291i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.891 + 0.453i)T \)
3 \( 1 + (-0.0637 - 0.997i)T \)
5 \( 1 + (0.924 - 0.380i)T \)
7 \( 1 + (-0.0318 + 0.999i)T \)
11 \( 1 + (-0.819 + 0.572i)T \)
13 \( 1 + (-0.410 + 0.912i)T \)
17 \( 1 + (0.666 - 0.745i)T \)
19 \( 1 + (0.0398 - 0.999i)T \)
23 \( 1 + (-0.442 + 0.896i)T \)
29 \( 1 + (-0.950 + 0.309i)T \)
31 \( 1 + (-0.505 + 0.862i)T \)
37 \( 1 + (-0.182 + 0.983i)T \)
41 \( 1 + (0.874 - 0.484i)T \)
43 \( 1 + (0.0995 + 0.995i)T \)
47 \( 1 + (0.162 + 0.986i)T \)
53 \( 1 + (0.885 + 0.463i)T \)
59 \( 1 + (-0.543 + 0.839i)T \)
61 \( 1 + (0.979 - 0.201i)T \)
67 \( 1 + (0.927 - 0.373i)T \)
71 \( 1 + (0.502 + 0.864i)T \)
73 \( 1 + (-0.941 + 0.336i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (0.901 - 0.431i)T \)
89 \( 1 + (0.996 + 0.0836i)T \)
97 \( 1 + (-0.999 - 0.0318i)T \)
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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

−17.572914743269177726838695498755, −16.999014730587967398641422470533, −16.56162919184362091074700681845, −15.99218363692337782402190780631, −14.9444074049923330823328788224, −14.53219810814700746132729269851, −13.58046131781996335377585011543, −12.93725041375511814439234174811, −12.18851487662248772630219942934, −11.05417654958925151545120477569, −10.69116445355830225397918224265, −10.17722833976273368756687793394, −9.82597498264480350157817920246, −8.997680683912962434603450795278, −8.02040881432745399868002783007, −7.73654856164524349480157793301, −6.60857120742050370253647113365, −5.78077863899646470114656492131, −5.28868155782445193559703157833, −3.8520512168321592366240730693, −3.65681161171408748173866334573, −2.63138382409106564019256864325, −2.01970921943837571596605499561, −0.76477703131809413414085577487, −0.11336457672801702911736427224, 1.03373580505260974757299908707, 1.75448743401284802050527092260, 2.37599696891821051680103047828, 2.903863317323707571947906656518, 4.84534549583845753059787793588, 5.35711222206285951448697238215, 5.88694966130212930584235347595, 6.7224347032675767784423958392, 7.30366932819074047716503802972, 7.94988152290513812197549060411, 8.85699344781317325213951732144, 9.299591933889148149069643128164, 9.81222565501098883586958307314, 10.84664158232209579966160750500, 11.59192944701645914880732710076, 12.20564129678952695002942550119, 12.89738770590174942106271768819, 13.67630315977038118984379025864, 14.31554830137687871771245696065, 14.947137357837515394684182122704, 15.863363921931550984958877309303, 16.40240177107347968107755633031, 17.19191269849498801945182562198, 17.826928738878174904779629092442, 18.11793656254426654455371813858

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