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Properties

Label 1-23e2-529.179-r0-0-0
Degree $1$
Conductor $529$
Sign $0.999 + 0.0118i$
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  + (0.481 − 0.876i)2-s + (−0.977 + 0.209i)3-s + (−0.535 − 0.844i)4-s + (0.566 + 0.824i)5-s + (−0.287 + 0.957i)6-s + (0.751 − 0.659i)7-s + (−0.998 + 0.0620i)8-s + (0.912 − 0.409i)9-s + (0.995 − 0.0991i)10-s + (0.227 + 0.973i)11-s + (0.700 + 0.713i)12-s + (−0.239 + 0.970i)13-s + (−0.215 − 0.976i)14-s + (−0.726 − 0.687i)15-s + (−0.426 + 0.904i)16-s + (−0.982 + 0.185i)17-s + ⋯
L(s)  = 1  + (0.481 − 0.876i)2-s + (−0.977 + 0.209i)3-s + (−0.535 − 0.844i)4-s + (0.566 + 0.824i)5-s + (−0.287 + 0.957i)6-s + (0.751 − 0.659i)7-s + (−0.998 + 0.0620i)8-s + (0.912 − 0.409i)9-s + (0.995 − 0.0991i)10-s + (0.227 + 0.973i)11-s + (0.700 + 0.713i)12-s + (−0.239 + 0.970i)13-s + (−0.215 − 0.976i)14-s + (−0.726 − 0.687i)15-s + (−0.426 + 0.904i)16-s + (−0.982 + 0.185i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.999 + 0.0118i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.999 + 0.0118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.309710022 + 0.007778116745i\)
\(L(\frac12)\) \(\approx\) \(1.309710022 + 0.007778116745i\)
\(L(1)\) \(\approx\) \(1.070008594 - 0.2148172110i\)
\(L(1)\) \(\approx\) \(1.070008594 - 0.2148172110i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.481 - 0.876i)T \)
3 \( 1 + (-0.977 + 0.209i)T \)
5 \( 1 + (0.566 + 0.824i)T \)
7 \( 1 + (0.751 - 0.659i)T \)
11 \( 1 + (0.227 + 0.973i)T \)
13 \( 1 + (-0.239 + 0.970i)T \)
17 \( 1 + (-0.982 + 0.185i)T \)
19 \( 1 + (0.879 - 0.476i)T \)
29 \( 1 + (0.717 + 0.696i)T \)
31 \( 1 + (-0.635 + 0.771i)T \)
37 \( 1 + (-0.999 + 0.0124i)T \)
41 \( 1 + (-0.691 + 0.722i)T \)
43 \( 1 + (0.524 + 0.851i)T \)
47 \( 1 + (0.854 - 0.519i)T \)
53 \( 1 + (0.995 + 0.0991i)T \)
59 \( 1 + (0.798 - 0.601i)T \)
61 \( 1 + (0.735 - 0.678i)T \)
67 \( 1 + (-0.791 + 0.611i)T \)
71 \( 1 + (-0.0929 + 0.995i)T \)
73 \( 1 + (0.717 - 0.696i)T \)
79 \( 1 + (0.901 + 0.432i)T \)
83 \( 1 + (0.922 + 0.386i)T \)
89 \( 1 + (0.626 - 0.779i)T \)
97 \( 1 + (-0.191 - 0.981i)T \)
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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

−23.77925692803640830869290970701, −22.46069858124364906495309101416, −22.150114447019433517940899882089, −21.24386749769647522476706938168, −20.50592471312161260158310356250, −18.92722415327510459056069195258, −17.89392696716873288321297495154, −17.572257563009664695743529937422, −16.68752822068322667158415879468, −15.92549945329869571218356224729, −15.185676899819866879357966550155, −13.86011930233160136146276749624, −13.34101556404509632715488470858, −12.2534974570120997832005749159, −11.799630544398911335597460851937, −10.55878763366897894920758384038, −9.23614395078419208037462116645, −8.417828538582086314947910036184, −7.51671709635848292586345181329, −6.27105899922596058830035766792, −5.50438177720341826026280298330, −5.136643069457304022121663044618, −3.97240328374901045740510770822, −2.30373000168976406211152386541, −0.7445168069821778260342811408, 1.33965351428374770079487491443, 2.17795747091238374493880852043, 3.68620103105774372235341878105, 4.63198260837661371025792686351, 5.26989068456890059106568443143, 6.639184461660682872278675851684, 7.081077206454189689463992714012, 9.05444019395535562324002004336, 9.95688266766218256888769256462, 10.6278208756983414028670297925, 11.35105148301002635320181586191, 12.01841418774259389746375904657, 13.12895067760345949326720476742, 14.03285553263336578370354390084, 14.71416835682683229812487664306, 15.6721097499374488799706273859, 17.02228820017514943371997534588, 17.92271142679375126404920188321, 18.08495262455882623026247571145, 19.38832404209075203179806536272, 20.30480560591881392300747797702, 21.19448328753024309865732022944, 21.91498151204782003018000786645, 22.41490394111969936153611812337, 23.34698531739600297487278161625

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