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Properties

Label 1-23e2-529.119-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.517 + 0.855i$
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  + (−0.952 + 0.305i)2-s + (0.999 − 0.0248i)3-s + (0.813 − 0.581i)4-s + (−0.381 + 0.924i)5-s + (−0.944 + 0.329i)6-s + (−0.191 + 0.981i)7-s + (−0.596 + 0.802i)8-s + (0.998 − 0.0496i)9-s + (0.0806 − 0.996i)10-s + (0.299 + 0.954i)11-s + (0.798 − 0.601i)12-s + (−0.471 + 0.882i)13-s + (−0.117 − 0.993i)14-s + (−0.358 + 0.933i)15-s + (0.323 − 0.946i)16-s + (0.940 + 0.340i)17-s + ⋯
L(s)  = 1  + (−0.952 + 0.305i)2-s + (0.999 − 0.0248i)3-s + (0.813 − 0.581i)4-s + (−0.381 + 0.924i)5-s + (−0.944 + 0.329i)6-s + (−0.191 + 0.981i)7-s + (−0.596 + 0.802i)8-s + (0.998 − 0.0496i)9-s + (0.0806 − 0.996i)10-s + (0.299 + 0.954i)11-s + (0.798 − 0.601i)12-s + (−0.471 + 0.882i)13-s + (−0.117 − 0.993i)14-s + (−0.358 + 0.933i)15-s + (0.323 − 0.946i)16-s + (0.940 + 0.340i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.517 + 0.855i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.517 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5440829325 + 0.9655144562i\)
\(L(\frac12)\) \(\approx\) \(0.5440829325 + 0.9655144562i\)
\(L(1)\) \(\approx\) \(0.8030909890 + 0.4609370757i\)
\(L(1)\) \(\approx\) \(0.8030909890 + 0.4609370757i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.952 + 0.305i)T \)
3 \( 1 + (0.999 - 0.0248i)T \)
5 \( 1 + (-0.381 + 0.924i)T \)
7 \( 1 + (-0.191 + 0.981i)T \)
11 \( 1 + (0.299 + 0.954i)T \)
13 \( 1 + (-0.471 + 0.882i)T \)
17 \( 1 + (0.940 + 0.340i)T \)
19 \( 1 + (0.392 - 0.919i)T \)
29 \( 1 + (0.524 - 0.851i)T \)
31 \( 1 + (-0.791 + 0.611i)T \)
37 \( 1 + (-0.982 + 0.185i)T \)
41 \( 1 + (-0.896 - 0.443i)T \)
43 \( 1 + (-0.907 + 0.421i)T \)
47 \( 1 + (-0.334 - 0.942i)T \)
53 \( 1 + (0.0806 + 0.996i)T \)
59 \( 1 + (-0.966 + 0.257i)T \)
61 \( 1 + (0.179 + 0.983i)T \)
67 \( 1 + (0.901 - 0.432i)T \)
71 \( 1 + (0.984 - 0.172i)T \)
73 \( 1 + (0.524 + 0.851i)T \)
79 \( 1 + (0.912 + 0.409i)T \)
83 \( 1 + (0.948 - 0.317i)T \)
89 \( 1 + (0.664 - 0.747i)T \)
97 \( 1 + (0.251 - 0.967i)T \)
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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

−23.46385435803739879579488703735, −22.08329874627494936770414999600, −21.01591640579444318968804919991, −20.441788671047422274657681647830, −19.88449310939712909659420726318, −19.20343135776254574248397524268, −18.381348613137051266557619922777, −17.14638580987179389538038580620, −16.45237380035491743820298292190, −15.89475429217781977275727729906, −14.70493791493975948664221233956, −13.70449381087567264779745734055, −12.771873727672687775838143024891, −12.04266353882885895912427957837, −10.78358919361645210026728499795, −9.9218038206760954814846216829, −9.23006670261371252398554322471, −8.06910475807492618312570127852, −7.90056610683284521255954188200, −6.75189792107482047473484885836, −5.17085853287091023086671098184, −3.604783678700333654115769042895, −3.32969742984015026817024660311, −1.64810223357981932895711505482, −0.72588765030621453251786621802, 1.75921865224127149836810750585, 2.5092266790447911788129414339, 3.49765567524822801604994185335, 5.01669679363230417827792691871, 6.53525192674866482387213311344, 7.09575736308591183727640248325, 7.970707590580357608100791431322, 8.934665977741785445980070928205, 9.6582545828201368526531041549, 10.367217716676380918780980754355, 11.71214779947274900749664847706, 12.28512729857043926709301506260, 13.88821266214644786699088898230, 14.72219748454212701288691325197, 15.23303313530769634235971029037, 15.88584696230795164350514178611, 17.09811274246552746310898641470, 18.22384017069521173824648936307, 18.67625195926931389937171159184, 19.49655467605040991051860857432, 19.948277169799696681033703121424, 21.19711470593426755044971678907, 21.863078889391915111829929828131, 23.0912799374411056678027956052, 24.04844221023941524324277242199

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