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Properties

Label 1-23e2-529.233-r0-0-0
Degree $1$
Conductor $529$
Sign $0.857 - 0.514i$
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  + (−0.860 − 0.508i)2-s + (0.105 − 0.994i)3-s + (0.481 + 0.876i)4-s + (−0.885 + 0.465i)5-s + (−0.596 + 0.802i)6-s + (−0.935 − 0.352i)7-s + (0.0310 − 0.999i)8-s + (−0.977 − 0.209i)9-s + (0.998 + 0.0496i)10-s + (0.783 − 0.621i)11-s + (0.922 − 0.386i)12-s + (−0.616 + 0.787i)13-s + (0.626 + 0.779i)14-s + (0.369 + 0.929i)15-s + (−0.535 + 0.844i)16-s + (−0.0929 + 0.995i)17-s + ⋯
L(s)  = 1  + (−0.860 − 0.508i)2-s + (0.105 − 0.994i)3-s + (0.481 + 0.876i)4-s + (−0.885 + 0.465i)5-s + (−0.596 + 0.802i)6-s + (−0.935 − 0.352i)7-s + (0.0310 − 0.999i)8-s + (−0.977 − 0.209i)9-s + (0.998 + 0.0496i)10-s + (0.783 − 0.621i)11-s + (0.922 − 0.386i)12-s + (−0.616 + 0.787i)13-s + (0.626 + 0.779i)14-s + (0.369 + 0.929i)15-s + (−0.535 + 0.844i)16-s + (−0.0929 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.857 - 0.514i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.857 - 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5382037413 - 0.1489190391i\)
\(L(\frac12)\) \(\approx\) \(0.5382037413 - 0.1489190391i\)
\(L(1)\) \(\approx\) \(0.5288817964 - 0.2025516356i\)
\(L(1)\) \(\approx\) \(0.5288817964 - 0.2025516356i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.860 - 0.508i)T \)
3 \( 1 + (0.105 - 0.994i)T \)
5 \( 1 + (-0.885 + 0.465i)T \)
7 \( 1 + (-0.935 - 0.352i)T \)
11 \( 1 + (0.783 - 0.621i)T \)
13 \( 1 + (-0.616 + 0.787i)T \)
17 \( 1 + (-0.0929 + 0.995i)T \)
19 \( 1 + (0.969 + 0.245i)T \)
29 \( 1 + (-0.926 + 0.375i)T \)
31 \( 1 + (-0.426 + 0.904i)T \)
37 \( 1 + (0.00620 - 0.999i)T \)
41 \( 1 + (0.392 - 0.919i)T \)
43 \( 1 + (-0.873 + 0.487i)T \)
47 \( 1 + (0.962 + 0.269i)T \)
53 \( 1 + (0.998 - 0.0496i)T \)
59 \( 1 + (0.948 + 0.317i)T \)
61 \( 1 + (0.931 + 0.363i)T \)
67 \( 1 + (0.323 - 0.946i)T \)
71 \( 1 + (-0.673 + 0.739i)T \)
73 \( 1 + (-0.926 - 0.375i)T \)
79 \( 1 + (0.975 - 0.221i)T \)
83 \( 1 + (0.980 - 0.197i)T \)
89 \( 1 + (0.901 + 0.432i)T \)
97 \( 1 + (-0.635 - 0.771i)T \)
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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

−23.555945815863517006766625767616, −22.56104830025550666950874744545, −22.26138441079293996879774730691, −20.45275405001815676897670730777, −20.239198443998618652489983452501, −19.47916322486250010673010205887, −18.54960005676723285315170241463, −17.37603183634330519141034435747, −16.59767540384749579650538183447, −16.01163644913700697838927241506, −15.22471606232539379158266882187, −14.77771826239965388903614008694, −13.3681384773558334525336605765, −11.94180779937132830348721385606, −11.44576978212318548418600567064, −10.09174859091357858794177498729, −9.51847424899935616561119646296, −8.88608457949312522031335367794, −7.784385818953719086825061700490, −6.95830023941367875386872963592, −5.6362171456702756174580412124, −4.856621486627543082535108007428, −3.62980117769240735468892710304, −2.53013646963528620044549603267, −0.5643709014238021496108151257, 0.828312529815802718746736872804, 2.094377829966815676173973251212, 3.315243319128989738127043088440, 3.844474702089080859834090492397, 6.03386427549242334676605046657, 7.03964322250374058698353717894, 7.36067057539704710062338938526, 8.55750353181354168758460888791, 9.266029746575359162390041842385, 10.486964280668090586065980630996, 11.38729217071411546999472965934, 12.09411086261854484809417624287, 12.77101731124579924115591428226, 13.88086978787979631294862479612, 14.82664305826513190880657606249, 16.19460568986441340148255658412, 16.685072667299391193152973582248, 17.69410526471896560452555564716, 18.64252850894638262482019238002, 19.4001499730593700142821819106, 19.52318437220753214955708540947, 20.427346633089143620344565937673, 21.88737110502765977736342636429, 22.44519765315948966061499562958, 23.51517693023457387239142987689

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