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Properties

Label 1-23e2-529.445-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: $\chi_{529} (445, \cdot )$
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

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

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