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Properties

Label 1-23e2-529.236-r0-0-0
Degree $1$
Conductor $529$
Sign $0.999 - 0.0313i$
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  + (−0.999 − 0.0372i)2-s + (0.969 − 0.245i)3-s + (0.997 + 0.0744i)4-s + (0.717 + 0.696i)5-s + (−0.977 + 0.209i)6-s + (0.346 − 0.938i)7-s + (−0.993 − 0.111i)8-s + (0.879 − 0.476i)9-s + (−0.691 − 0.722i)10-s + (0.995 + 0.0991i)11-s + (0.984 − 0.172i)12-s + (−0.191 + 0.981i)13-s + (−0.381 + 0.924i)14-s + (0.867 + 0.498i)15-s + (0.988 + 0.148i)16-s + (−0.944 − 0.329i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0372i)2-s + (0.969 − 0.245i)3-s + (0.997 + 0.0744i)4-s + (0.717 + 0.696i)5-s + (−0.977 + 0.209i)6-s + (0.346 − 0.938i)7-s + (−0.993 − 0.111i)8-s + (0.879 − 0.476i)9-s + (−0.691 − 0.722i)10-s + (0.995 + 0.0991i)11-s + (0.984 − 0.172i)12-s + (−0.191 + 0.981i)13-s + (−0.381 + 0.924i)14-s + (0.867 + 0.498i)15-s + (0.988 + 0.148i)16-s + (−0.944 − 0.329i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.999 - 0.0313i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.999 - 0.0313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.580782925 + 0.02475182570i\)
\(L(\frac12)\) \(\approx\) \(1.580782925 + 0.02475182570i\)
\(L(1)\) \(\approx\) \(1.187596804 - 0.03231005134i\)
\(L(1)\) \(\approx\) \(1.187596804 - 0.03231005134i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.999 - 0.0372i)T \)
3 \( 1 + (0.969 - 0.245i)T \)
5 \( 1 + (0.717 + 0.696i)T \)
7 \( 1 + (0.346 - 0.938i)T \)
11 \( 1 + (0.995 + 0.0991i)T \)
13 \( 1 + (-0.191 + 0.981i)T \)
17 \( 1 + (-0.944 - 0.329i)T \)
19 \( 1 + (0.626 + 0.779i)T \)
29 \( 1 + (-0.726 + 0.687i)T \)
31 \( 1 + (0.955 - 0.293i)T \)
37 \( 1 + (-0.287 - 0.957i)T \)
41 \( 1 + (-0.117 - 0.993i)T \)
43 \( 1 + (-0.358 + 0.933i)T \)
47 \( 1 + (0.962 - 0.269i)T \)
53 \( 1 + (-0.691 + 0.722i)T \)
59 \( 1 + (-0.860 - 0.508i)T \)
61 \( 1 + (0.227 + 0.973i)T \)
67 \( 1 + (-0.239 + 0.970i)T \)
71 \( 1 + (-0.166 - 0.985i)T \)
73 \( 1 + (-0.726 - 0.687i)T \)
79 \( 1 + (-0.471 - 0.882i)T \)
83 \( 1 + (-0.996 + 0.0868i)T \)
89 \( 1 + (-0.556 - 0.831i)T \)
97 \( 1 + (0.827 - 0.561i)T \)
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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

−24.20691141763263176621339512213, −22.1687522236109561331887333640, −21.6636153378398912434200815351, −20.6519992437571912873232387159, −20.13434097760203931794713993963, −19.37831972167853538923208796611, −18.421546822966568214837798666, −17.57851512639249756622606856096, −16.92847978916608855474287785055, −15.62802932629214131226487169594, −15.32295732957399802627992324753, −14.233587518891776530755701904671, −13.20184099864792922367612790392, −12.23664649403269090721079464854, −11.22456948465989612998506262006, −10.00260945903976383450542451005, −9.37327690538947420224045142608, −8.66075641831175900382297967416, −8.11547176919037611072535247834, −6.819186056642910366353238531608, −5.76119909041781772107655509767, −4.643617037795064682111540747007, −3.069877264713615786715036915216, −2.18865216119835266888202195905, −1.26648653193792989726241710092, 1.38317319445025680822638894750, 2.02070796827655413570715263799, 3.20910054108646694919580604457, 4.23963684222143710100659910886, 6.159877613483047219239840804779, 7.05620985485376908772408259651, 7.44838454715642579306154441809, 8.79528121857930531550865569223, 9.41101088882115733345426399417, 10.197026048775733883152370477274, 11.12816105575395128964254065721, 12.09394675996994930032218579462, 13.45956973447564594177898921371, 14.23484302076912487954888747348, 14.74372812155987486215600400464, 15.94574967196579775449149755843, 16.96833287835900129334586293168, 17.65926094826076678958087042479, 18.50141137536633021210138050536, 19.2207985979820150504027960527, 20.03010636786607364563455538676, 20.694216853713384694422193553365, 21.49242485966822275648115274292, 22.50586039562887239153508876475, 23.87703006769037741300256771630

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