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Properties

Label 1-4729-4729.1055-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.517 - 0.855i$
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  + (−0.944 + 0.328i)2-s + (−0.0875 + 0.996i)3-s + (0.783 − 0.620i)4-s + (−0.951 + 0.308i)5-s + (−0.244 − 0.969i)6-s + (0.300 + 0.953i)7-s + (−0.536 + 0.843i)8-s + (−0.984 − 0.174i)9-s + (0.796 − 0.604i)10-s + (0.150 − 0.988i)11-s + (0.549 + 0.835i)12-s + (0.862 + 0.506i)13-s + (−0.597 − 0.801i)14-s + (−0.224 − 0.974i)15-s + (0.229 − 0.973i)16-s + (−0.919 − 0.393i)17-s + ⋯
L(s)  = 1  + (−0.944 + 0.328i)2-s + (−0.0875 + 0.996i)3-s + (0.783 − 0.620i)4-s + (−0.951 + 0.308i)5-s + (−0.244 − 0.969i)6-s + (0.300 + 0.953i)7-s + (−0.536 + 0.843i)8-s + (−0.984 − 0.174i)9-s + (0.796 − 0.604i)10-s + (0.150 − 0.988i)11-s + (0.549 + 0.835i)12-s + (0.862 + 0.506i)13-s + (−0.597 − 0.801i)14-s + (−0.224 − 0.974i)15-s + (0.229 − 0.973i)16-s + (−0.919 − 0.393i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.517 - 0.855i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.517 - 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04943477001 + 0.02787207648i\)
\(L(\frac12)\) \(\approx\) \(0.04943477001 + 0.02787207648i\)
\(L(1)\) \(\approx\) \(0.4282099120 + 0.2725582196i\)
\(L(1)\) \(\approx\) \(0.4282099120 + 0.2725582196i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.944 + 0.328i)T \)
3 \( 1 + (-0.0875 + 0.996i)T \)
5 \( 1 + (-0.951 + 0.308i)T \)
7 \( 1 + (0.300 + 0.953i)T \)
11 \( 1 + (0.150 - 0.988i)T \)
13 \( 1 + (0.862 + 0.506i)T \)
17 \( 1 + (-0.919 - 0.393i)T \)
19 \( 1 + (-0.490 - 0.871i)T \)
23 \( 1 + (-0.601 + 0.798i)T \)
29 \( 1 + (0.944 + 0.328i)T \)
31 \( 1 + (-0.939 - 0.343i)T \)
37 \( 1 + (0.892 + 0.450i)T \)
41 \( 1 + (0.563 - 0.826i)T \)
43 \( 1 + (-0.476 + 0.879i)T \)
47 \( 1 + (-0.0610 + 0.998i)T \)
53 \( 1 + (-0.877 + 0.479i)T \)
59 \( 1 + (-0.254 + 0.966i)T \)
61 \( 1 + (-0.0504 + 0.998i)T \)
67 \( 1 + (-0.00265 + 0.999i)T \)
71 \( 1 + (0.558 - 0.829i)T \)
73 \( 1 + (-0.0132 + 0.999i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.995 - 0.0902i)T \)
89 \( 1 + (-0.0823 - 0.996i)T \)
97 \( 1 + (0.300 + 0.953i)T \)
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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

−18.268956230045603660306184854890, −17.71803731149385471743218685156, −17.09524541789925617974661315303, −16.490603454648939936623384054936, −15.82021615476036017116581139474, −14.97594356578063262095518603945, −14.29750205344452633982203746545, −13.20809229853972762041997351157, −12.72614040463713556655179771369, −12.19799881982833009353020291616, −11.432096742317285648431728264642, −10.83783374238929765448672808755, −10.34157449430708237591519275291, −9.26778812396667998188729796724, −8.36823372651066112291866833411, −8.06701314092698117361610469557, −7.50580476635414893450454061660, −6.65621465003279240796488875679, −6.30687695660620476293297878337, −4.90204966356082588281296671615, −4.000192417027473942572726648442, −3.45816302014872584658526778543, −2.25099010856733408247777225932, −1.63271134699294968450938638510, −0.82166804544045307810592481255, 0.02846398442255321381461828839, 1.25991425305269051002424820170, 2.54973844894915597516973794084, 3.01815561379992570243815276343, 4.06866536299639325868406909084, 4.747371518180113466567789026743, 5.77061160243377690025046361876, 6.208336845075328914172181847818, 7.061782377294142431886756035, 8.08400513533690002317987520033, 8.59232803249824623236366175792, 9.02525624030214680104335806139, 9.6782548593643606184565743201, 10.81350540016808575977731559526, 11.22825937740176274511370647257, 11.42219124945497091650362374163, 12.29057528966777152814666776302, 13.60471037479824779284684323037, 14.39194270485498811110118352864, 14.99702870319038590648513101507, 15.674288817944152026422060910002, 15.94948061634619334014236954974, 16.46486380714632366950751378516, 17.403725242427937206884662412383, 18.086589621848547147102903748896

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