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Properties

Label 1-23e2-529.279-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.855 + 0.518i$
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  + (0.0806 + 0.996i)2-s + (−0.873 + 0.487i)3-s + (−0.986 + 0.160i)4-s + (0.813 − 0.581i)5-s + (−0.556 − 0.831i)6-s + (−0.0186 + 0.999i)7-s + (−0.239 − 0.970i)8-s + (0.524 − 0.851i)9-s + (0.645 + 0.763i)10-s + (−0.673 − 0.739i)11-s + (0.783 − 0.621i)12-s + (0.154 + 0.987i)13-s + (−0.998 + 0.0620i)14-s + (−0.426 + 0.904i)15-s + (0.948 − 0.317i)16-s + (0.664 + 0.747i)17-s + ⋯
L(s)  = 1  + (0.0806 + 0.996i)2-s + (−0.873 + 0.487i)3-s + (−0.986 + 0.160i)4-s + (0.813 − 0.581i)5-s + (−0.556 − 0.831i)6-s + (−0.0186 + 0.999i)7-s + (−0.239 − 0.970i)8-s + (0.524 − 0.851i)9-s + (0.645 + 0.763i)10-s + (−0.673 − 0.739i)11-s + (0.783 − 0.621i)12-s + (0.154 + 0.987i)13-s + (−0.998 + 0.0620i)14-s + (−0.426 + 0.904i)15-s + (0.948 − 0.317i)16-s + (0.664 + 0.747i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.855 + 0.518i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.855 + 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2537947196 + 0.9086369944i\)
\(L(\frac12)\) \(\approx\) \(0.2537947196 + 0.9086369944i\)
\(L(1)\) \(\approx\) \(0.6319519487 + 0.5652358403i\)
\(L(1)\) \(\approx\) \(0.6319519487 + 0.5652358403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.0806 + 0.996i)T \)
3 \( 1 + (-0.873 + 0.487i)T \)
5 \( 1 + (0.813 - 0.581i)T \)
7 \( 1 + (-0.0186 + 0.999i)T \)
11 \( 1 + (-0.673 - 0.739i)T \)
13 \( 1 + (0.154 + 0.987i)T \)
17 \( 1 + (0.664 + 0.747i)T \)
19 \( 1 + (-0.358 - 0.933i)T \)
29 \( 1 + (0.437 + 0.899i)T \)
31 \( 1 + (0.798 + 0.601i)T \)
37 \( 1 + (0.626 + 0.779i)T \)
41 \( 1 + (0.00620 + 0.999i)T \)
43 \( 1 + (-0.492 + 0.870i)T \)
47 \( 1 + (-0.0682 + 0.997i)T \)
53 \( 1 + (0.645 - 0.763i)T \)
59 \( 1 + (-0.806 + 0.591i)T \)
61 \( 1 + (-0.972 - 0.233i)T \)
67 \( 1 + (-0.966 - 0.257i)T \)
71 \( 1 + (0.912 - 0.409i)T \)
73 \( 1 + (0.437 - 0.899i)T \)
79 \( 1 + (0.717 - 0.696i)T \)
83 \( 1 + (-0.944 + 0.329i)T \)
89 \( 1 + (0.0310 + 0.999i)T \)
97 \( 1 + (0.275 + 0.961i)T \)
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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

−22.922230814670080081320403838773, −22.572971323980155635290987659120, −21.34378287205049627994558792561, −20.76553764850653086888757837327, −19.83935329556230458095850794027, −18.6654508998914766486514986685, −18.2584802277262407451658924110, −17.377585707769659812213062196363, −16.88348854713977672445548096722, −15.38168526046171347276662970191, −14.136266737723792347793142344538, −13.50018528694730113376909011168, −12.77692023651913417172849013999, −11.92480607552392431201851217547, −10.81594463227111495278715510039, −10.26730366870186415046077850909, −9.80122013502880286081638601318, −8.025013631734418521826924051438, −7.234909493088320105280336368778, −5.96260027642734201915541862394, −5.23295797644703571952061501677, −4.12006674790289206535394654179, −2.80106999458818659598285019083, −1.8129213558884010022512968687, −0.62710494527697155585117321137, 1.25048228823993936039870383944, 3.06606703977354811736449886962, 4.61322187906232101203605218511, 5.1118393980247376631307433409, 6.13366702427030231327033574615, 6.45910483438181481789274604446, 8.14948296039661030329698087570, 8.97231477998178333907459113729, 9.64787501203729550964439087635, 10.70096542061601461726039909753, 11.94615219573503434238982222874, 12.75671396620425794858186051343, 13.57609013392656655807518520545, 14.66814023360833529867359383447, 15.51786401926754432264966987586, 16.391661204902366146985197826617, 16.74773770119516750630259053946, 17.84510515782777320890299616271, 18.307134659582604622402800552286, 19.33296697194032152273806526122, 21.19037887713739885128450136809, 21.46695847202401028108909877088, 21.997094988560056674431984933159, 23.18533401586430980340708410296, 23.96023164110891967707510926236

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