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Properties

Label 1-23e2-529.47-r0-0-0
Degree $1$
Conductor $529$
Sign $0.878 - 0.478i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
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  + (0.962 − 0.269i)2-s + (0.962 − 0.269i)3-s + (0.854 − 0.519i)4-s + (−0.917 + 0.398i)5-s + (0.854 − 0.519i)6-s + (0.854 + 0.519i)7-s + (0.682 − 0.730i)8-s + (0.854 − 0.519i)9-s + (−0.775 + 0.631i)10-s + (0.203 + 0.979i)11-s + (0.682 − 0.730i)12-s + (−0.775 − 0.631i)13-s + (0.962 + 0.269i)14-s + (−0.775 + 0.631i)15-s + (0.460 − 0.887i)16-s + (−0.775 − 0.631i)17-s + ⋯
L(s)  = 1  + (0.962 − 0.269i)2-s + (0.962 − 0.269i)3-s + (0.854 − 0.519i)4-s + (−0.917 + 0.398i)5-s + (0.854 − 0.519i)6-s + (0.854 + 0.519i)7-s + (0.682 − 0.730i)8-s + (0.854 − 0.519i)9-s + (−0.775 + 0.631i)10-s + (0.203 + 0.979i)11-s + (0.682 − 0.730i)12-s + (−0.775 − 0.631i)13-s + (0.962 + 0.269i)14-s + (−0.775 + 0.631i)15-s + (0.460 − 0.887i)16-s + (−0.775 − 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.878 - 0.478i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.878 - 0.478i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.205685697 - 0.8165896653i\)
\(L(\frac12)\) \(\approx\) \(3.205685697 - 0.8165896653i\)
\(L(1)\) \(\approx\) \(2.312312158 - 0.4344563895i\)
\(L(1)\) \(\approx\) \(2.312312158 - 0.4344563895i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.962 - 0.269i)T \)
3 \( 1 + (0.962 - 0.269i)T \)
5 \( 1 + (-0.917 + 0.398i)T \)
7 \( 1 + (0.854 + 0.519i)T \)
11 \( 1 + (0.203 + 0.979i)T \)
13 \( 1 + (-0.775 - 0.631i)T \)
17 \( 1 + (-0.775 - 0.631i)T \)
19 \( 1 + (0.962 - 0.269i)T \)
29 \( 1 + (0.203 + 0.979i)T \)
31 \( 1 + (-0.576 + 0.816i)T \)
37 \( 1 + (0.460 + 0.887i)T \)
41 \( 1 + (-0.334 + 0.942i)T \)
43 \( 1 + (-0.0682 - 0.997i)T \)
47 \( 1 + (-0.576 - 0.816i)T \)
53 \( 1 + (-0.775 - 0.631i)T \)
59 \( 1 + (0.962 + 0.269i)T \)
61 \( 1 + (-0.917 + 0.398i)T \)
67 \( 1 + (0.203 + 0.979i)T \)
71 \( 1 + (-0.334 - 0.942i)T \)
73 \( 1 + (0.203 - 0.979i)T \)
79 \( 1 + (-0.0682 - 0.997i)T \)
83 \( 1 + (-0.917 + 0.398i)T \)
89 \( 1 + (-0.990 - 0.136i)T \)
97 \( 1 + (-0.334 - 0.942i)T \)
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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

−23.87104194950127393440532494879, −22.70472199526372681989213199051, −21.74218335409519004364238591717, −21.108357932478382922123706928178, −20.237296743043541910833983587464, −19.72467016544490722672015278933, −18.85652247777317650708761866386, −17.289859879960804305245659280114, −16.440375781290035404119286540199, −15.741602789222134609697379345297, −14.85713798708235232940685812278, −14.2265525927424357345658368050, −13.50455181437983795290672818599, −12.53562279767309532965594063353, −11.4727084999523402529736850515, −10.916763545652152269731356191621, −9.412247595992014136937259405004, −8.24280463224982112521992777972, −7.80119852624542958944686571105, −6.878529115563087057402936342564, −5.394022936335828976391337773414, −4.28988147172846234840182397601, −3.97523789269519345262727625435, −2.78154264202211578200415776396, −1.54945324817881846286663482547, 1.49330705405257087343013675190, 2.57121699957725734424026528995, 3.275232629088289901086628303110, 4.49219206883336513578350483036, 5.1007145040430908339069985278, 6.863912555013399333262407344405, 7.32119560711809956866785983641, 8.28317450189534390351614140044, 9.500578405100377941772181699965, 10.55273325896680080463749794647, 11.69641824768897169751038025836, 12.1802058909406825750140569461, 13.12172923540446113415599350818, 14.18690701624146741344206835308, 14.88028495094182585385827483691, 15.256602710999176022708256842761, 16.13537881647713266307320174498, 17.86553985541654534647635118806, 18.4782030053619587564509549384, 19.69320426036837704703693446849, 20.05813962652660226389591055339, 20.701390475025530588032258490561, 21.91174224334812732543978914875, 22.413999358278820787424191039529, 23.54643422628939857192540051531

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