Properties

Label 1-4729-4729.2058-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.777 - 0.629i$
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  + (0.880 − 0.474i)2-s + (0.275 − 0.961i)3-s + (0.549 − 0.835i)4-s + (0.944 + 0.328i)5-s + (−0.213 − 0.976i)6-s + (−0.601 + 0.798i)7-s + (0.0875 − 0.996i)8-s + (−0.848 − 0.529i)9-s + (0.987 − 0.158i)10-s + (0.930 − 0.366i)11-s + (−0.651 − 0.758i)12-s + (−0.305 + 0.952i)13-s + (−0.150 + 0.988i)14-s + (0.576 − 0.817i)15-s + (−0.395 − 0.918i)16-s + (−0.0478 + 0.998i)17-s + ⋯
L(s)  = 1  + (0.880 − 0.474i)2-s + (0.275 − 0.961i)3-s + (0.549 − 0.835i)4-s + (0.944 + 0.328i)5-s + (−0.213 − 0.976i)6-s + (−0.601 + 0.798i)7-s + (0.0875 − 0.996i)8-s + (−0.848 − 0.529i)9-s + (0.987 − 0.158i)10-s + (0.930 − 0.366i)11-s + (−0.651 − 0.758i)12-s + (−0.305 + 0.952i)13-s + (−0.150 + 0.988i)14-s + (0.576 − 0.817i)15-s + (−0.395 − 0.918i)16-s + (−0.0478 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.777 - 0.629i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (2058, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.777 - 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.899641498 - 1.380401276i\)
\(L(\frac12)\) \(\approx\) \(3.899641498 - 1.380401276i\)
\(L(1)\) \(\approx\) \(2.040441634 - 0.8386937816i\)
\(L(1)\) \(\approx\) \(2.040441634 - 0.8386937816i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.880 - 0.474i)T \)
3 \( 1 + (0.275 - 0.961i)T \)
5 \( 1 + (0.944 + 0.328i)T \)
7 \( 1 + (-0.601 + 0.798i)T \)
11 \( 1 + (0.930 - 0.366i)T \)
13 \( 1 + (-0.305 + 0.952i)T \)
17 \( 1 + (-0.0478 + 0.998i)T \)
19 \( 1 + (0.915 + 0.402i)T \)
23 \( 1 + (0.998 + 0.0557i)T \)
29 \( 1 + (-0.474 + 0.880i)T \)
31 \( 1 + (-0.898 + 0.439i)T \)
37 \( 1 + (0.821 - 0.569i)T \)
41 \( 1 + (0.236 + 0.971i)T \)
43 \( 1 + (0.860 + 0.509i)T \)
47 \( 1 + (-0.726 + 0.687i)T \)
53 \( 1 + (-0.595 - 0.803i)T \)
59 \( 1 + (0.589 - 0.808i)T \)
61 \( 1 + (-0.388 + 0.921i)T \)
67 \( 1 + (0.103 - 0.994i)T \)
71 \( 1 + (-0.901 + 0.431i)T \)
73 \( 1 + (-0.495 + 0.868i)T \)
79 \( 1 + iT \)
83 \( 1 + (0.927 + 0.373i)T \)
89 \( 1 + (0.997 - 0.0717i)T \)
97 \( 1 + (0.601 - 0.798i)T \)
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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

−17.698641524722151539210223541163, −17.39308263505264246521360720915, −16.64772722832214543373539652946, −16.302308336553361705803687467415, −15.48189808284760304382271043578, −14.81701077728243656216324410912, −14.266216408624057800155088405843, −13.469626954844096483368553105844, −13.287298980863488700044818748023, −12.31336373292458928063341101595, −11.50238772184572627712826778295, −10.75975534785955566103809169752, −9.99406207193653615057536012460, −9.32691298946158033604735435116, −8.91603346937917306026789638307, −7.64686475275452926419944182561, −7.22580158553054245270942787462, −6.24485990358664836150917851376, −5.616919537822577914241648052250, −4.90716350396994251391987873600, −4.3723799928590592488145921408, −3.46605649965029986672516090831, −2.93463581032578111792854053245, −2.11232607499628956152212757704, −0.75555545552785545172186688332, 1.209084534304439531001877390867, 1.66194598595514017471363389162, 2.43432767422881226578016060220, 3.15599597238467292404454949034, 3.707349337941620635973360919065, 4.96719980208510433007882196511, 5.76955201774514722797784152636, 6.23550933023998591661365403648, 6.74509163741715106733704261700, 7.45391937495083543330462052209, 8.76913649465146991964899027676, 9.321830874072039974741640140823, 9.76426445058484114031189528792, 11.00861724494323571182413188434, 11.39598015040104295618263632253, 12.24876590925033907365018551522, 12.83336872360316656412280068442, 13.19326856070802004671948915130, 14.115764452354076822827195037041, 14.59490380282720298064176556368, 14.81939512706893965241262586678, 16.14379674531944010483373795770, 16.66067689681405605186728553329, 17.57442696819344755499809561665, 18.36953207635183087039818047697

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