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Properties

Label 1-23e2-529.150-r0-0-0
Degree $1$
Conductor $529$
Sign $0.122 - 0.992i$
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  + (0.503 + 0.863i)2-s + (0.783 − 0.621i)3-s + (−0.492 + 0.870i)4-s + (−0.907 + 0.421i)5-s + (0.931 + 0.363i)6-s + (−0.885 − 0.465i)7-s + (−0.999 + 0.0124i)8-s + (0.227 − 0.973i)9-s + (−0.820 − 0.571i)10-s + (−0.935 + 0.352i)11-s + (0.154 + 0.987i)12-s + (0.626 − 0.779i)13-s + (−0.0434 − 0.999i)14-s + (−0.449 + 0.893i)15-s + (−0.514 − 0.857i)16-s + (−0.999 + 0.0372i)17-s + ⋯
L(s)  = 1  + (0.503 + 0.863i)2-s + (0.783 − 0.621i)3-s + (−0.492 + 0.870i)4-s + (−0.907 + 0.421i)5-s + (0.931 + 0.363i)6-s + (−0.885 − 0.465i)7-s + (−0.999 + 0.0124i)8-s + (0.227 − 0.973i)9-s + (−0.820 − 0.571i)10-s + (−0.935 + 0.352i)11-s + (0.154 + 0.987i)12-s + (0.626 − 0.779i)13-s + (−0.0434 − 0.999i)14-s + (−0.449 + 0.893i)15-s + (−0.514 − 0.857i)16-s + (−0.999 + 0.0372i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.122 - 0.992i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (150, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.122 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5462164547 - 0.4829346966i\)
\(L(\frac12)\) \(\approx\) \(0.5462164547 - 0.4829346966i\)
\(L(1)\) \(\approx\) \(0.9807578205 + 0.1304676208i\)
\(L(1)\) \(\approx\) \(0.9807578205 + 0.1304676208i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.503 + 0.863i)T \)
3 \( 1 + (0.783 - 0.621i)T \)
5 \( 1 + (-0.907 + 0.421i)T \)
7 \( 1 + (-0.885 - 0.465i)T \)
11 \( 1 + (-0.935 + 0.352i)T \)
13 \( 1 + (0.626 - 0.779i)T \)
17 \( 1 + (-0.999 + 0.0372i)T \)
19 \( 1 + (0.995 - 0.0991i)T \)
29 \( 1 + (-0.709 - 0.704i)T \)
31 \( 1 + (-0.471 - 0.882i)T \)
37 \( 1 + (-0.311 - 0.950i)T \)
41 \( 1 + (-0.986 + 0.160i)T \)
43 \( 1 + (-0.673 - 0.739i)T \)
47 \( 1 + (0.203 - 0.979i)T \)
53 \( 1 + (-0.820 + 0.571i)T \)
59 \( 1 + (-0.726 + 0.687i)T \)
61 \( 1 + (0.988 - 0.148i)T \)
67 \( 1 + (0.879 + 0.476i)T \)
71 \( 1 + (-0.0186 + 0.999i)T \)
73 \( 1 + (-0.709 + 0.704i)T \)
79 \( 1 + (0.392 - 0.919i)T \)
83 \( 1 + (-0.759 - 0.650i)T \)
89 \( 1 + (-0.691 + 0.722i)T \)
97 \( 1 + (-0.556 + 0.831i)T \)
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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

−23.64697613470425283589627600588, −22.53726815754624183228675563489, −21.97155955364930006232329246987, −21.03054081841863893630557227811, −20.33253715653899613743013486045, −19.713246231883444116860530383358, −18.89610940922296286838861060633, −18.340726441865742119777361935090, −16.43799985923378626100416279463, −15.78815836028721369231505129469, −15.28426042682693158914784702624, −14.10364717681829991659999643641, −13.29624924880712287922521788548, −12.63928987516901723739513611671, −11.49859747611289768474741294241, −10.81243174138100276128858995011, −9.69708275142611479750133488645, −8.97519866745485217975511684457, −8.25198727308639895325486501424, −6.80616663989287170286058396578, −5.373610692376471739456565439107, −4.595280312864226226613960736138, −3.478596112587786588100290287287, −3.06136344757831532889645184532, −1.70721780260159847158299125330, 0.282679358063670720114945121257, 2.55206993586771415736750965365, 3.42687691152941370787136312104, 4.08898447711563200936073463689, 5.57640420892006139393512758264, 6.71052845959719412043058130405, 7.35427379729533602789766986148, 7.98984035574719686913501592784, 8.92779290221178297786224551798, 10.073651104496210150178277143734, 11.406959820492048814836625049, 12.52107230121239509342875260268, 13.2110936905428426702987891230, 13.73004515344888375561525331817, 14.951860102236145354130993053395, 15.52637938604084177006866917566, 16.072827311478754953983372578485, 17.402386589195173560545170847768, 18.342861065047758898439166919423, 18.86568062944836881559250467300, 20.27410660792717981694031866744, 20.340361100747899222146108370943, 21.92515630237605714245323891058, 22.82031762271053836267213818476, 23.3211386863902744669394462267

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