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Properties

Label 1-23e2-529.370-r0-0-0
Degree $1$
Conductor $529$
Sign $0.618 + 0.785i$
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  + (0.717 − 0.696i)2-s + (0.586 + 0.809i)3-s + (0.0310 − 0.999i)4-s + (0.664 + 0.747i)5-s + (0.984 + 0.172i)6-s + (−0.514 + 0.857i)7-s + (−0.673 − 0.739i)8-s + (−0.311 + 0.950i)9-s + (0.997 + 0.0744i)10-s + (−0.535 + 0.844i)11-s + (0.827 − 0.561i)12-s + (−0.977 − 0.209i)13-s + (0.227 + 0.973i)14-s + (−0.215 + 0.976i)15-s + (−0.998 − 0.0620i)16-s + (0.798 − 0.601i)17-s + ⋯
L(s)  = 1  + (0.717 − 0.696i)2-s + (0.586 + 0.809i)3-s + (0.0310 − 0.999i)4-s + (0.664 + 0.747i)5-s + (0.984 + 0.172i)6-s + (−0.514 + 0.857i)7-s + (−0.673 − 0.739i)8-s + (−0.311 + 0.950i)9-s + (0.997 + 0.0744i)10-s + (−0.535 + 0.844i)11-s + (0.827 − 0.561i)12-s + (−0.977 − 0.209i)13-s + (0.227 + 0.973i)14-s + (−0.215 + 0.976i)15-s + (−0.998 − 0.0620i)16-s + (0.798 − 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.618 + 0.785i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (370, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.618 + 0.785i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.067192174 + 1.003375419i\)
\(L(\frac12)\) \(\approx\) \(2.067192174 + 1.003375419i\)
\(L(1)\) \(\approx\) \(1.738422934 + 0.2447841633i\)
\(L(1)\) \(\approx\) \(1.738422934 + 0.2447841633i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.717 - 0.696i)T \)
3 \( 1 + (0.586 + 0.809i)T \)
5 \( 1 + (0.664 + 0.747i)T \)
7 \( 1 + (-0.514 + 0.857i)T \)
11 \( 1 + (-0.535 + 0.844i)T \)
13 \( 1 + (-0.977 - 0.209i)T \)
17 \( 1 + (0.798 - 0.601i)T \)
19 \( 1 + (0.931 + 0.363i)T \)
29 \( 1 + (0.545 + 0.837i)T \)
31 \( 1 + (0.992 - 0.123i)T \)
37 \( 1 + (0.700 + 0.713i)T \)
41 \( 1 + (0.179 + 0.983i)T \)
43 \( 1 + (-0.691 - 0.722i)T \)
47 \( 1 + (-0.917 - 0.398i)T \)
53 \( 1 + (0.997 - 0.0744i)T \)
59 \( 1 + (-0.885 - 0.465i)T \)
61 \( 1 + (-0.847 - 0.530i)T \)
67 \( 1 + (-0.287 - 0.957i)T \)
71 \( 1 + (0.948 + 0.317i)T \)
73 \( 1 + (0.545 - 0.837i)T \)
79 \( 1 + (-0.944 + 0.329i)T \)
83 \( 1 + (0.955 - 0.293i)T \)
89 \( 1 + (0.783 + 0.621i)T \)
97 \( 1 + (0.969 - 0.245i)T \)
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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

−23.54498626173192409038877167379, −22.81799848339337604191866974237, −21.558989884535049811306261113506, −21.034494194449996256088460490757, −20.04716887487527864809832647891, −19.329223547331248804307214974079, −18.08452821761835211074303125331, −17.24548038384663339651728044893, −16.62750364936002878338925364113, −15.72922792521477200159553327131, −14.4934464551805422975707319937, −13.81801841938925041882030002348, −13.29946540089315791286640845697, −12.55663514898196848316618298830, −11.73089589733739132018044384648, −10.094411466767009441081104263438, −9.14723708110467907176193349941, −8.10114434297987641324196590666, −7.48178458636291313126286650452, −6.42221385043289143796534487178, −5.669247255290998544968243398215, −4.53162885524559750961518702444, −3.33906847531956292932287545378, −2.456594590414486594404338814130, −0.883015831790137363871578209049, 1.92780634162574991417009204472, 2.85234910482861782864230643038, 3.2266335084071377474561056368, 4.85368988168120399700839131698, 5.32857995120776664575470025887, 6.50823135052215164763462525431, 7.738604062898595065048019501764, 9.292917905163542607672502829760, 9.90782898642917867111076031509, 10.26730157355413759598500063955, 11.57848947995425061895745228987, 12.413327745462348256703184107527, 13.44505767251514265291097319326, 14.23930996387475274524677176149, 14.984790655901957937676773621562, 15.49664675419246378128159977772, 16.602541678572577583275113159275, 18.09149937035279392102070157500, 18.66023289987666814908036020028, 19.66921132254868178452137805953, 20.36884066109185601149056472198, 21.326937953115778656134708719409, 21.774255736857303120524815369544, 22.59642924705434926483319244260, 23.04900941650068621867714787881

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