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Properties

Label 1-23e2-529.58-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.515 + 0.857i$
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  + (−0.535 + 0.844i)2-s + (0.912 + 0.409i)3-s + (−0.426 − 0.904i)4-s + (−0.358 − 0.933i)5-s + (−0.834 + 0.551i)6-s + (0.130 + 0.991i)7-s + (0.992 + 0.123i)8-s + (0.664 + 0.747i)9-s + (0.980 + 0.197i)10-s + (−0.896 − 0.443i)11-s + (−0.0186 − 0.999i)12-s + (−0.885 + 0.465i)13-s + (−0.907 − 0.421i)14-s + (0.0558 − 0.998i)15-s + (−0.635 + 0.771i)16-s + (0.931 + 0.363i)17-s + ⋯
L(s)  = 1  + (−0.535 + 0.844i)2-s + (0.912 + 0.409i)3-s + (−0.426 − 0.904i)4-s + (−0.358 − 0.933i)5-s + (−0.834 + 0.551i)6-s + (0.130 + 0.991i)7-s + (0.992 + 0.123i)8-s + (0.664 + 0.747i)9-s + (0.980 + 0.197i)10-s + (−0.896 − 0.443i)11-s + (−0.0186 − 0.999i)12-s + (−0.885 + 0.465i)13-s + (−0.907 − 0.421i)14-s + (0.0558 − 0.998i)15-s + (−0.635 + 0.771i)16-s + (0.931 + 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.515 + 0.857i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.515 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5487724692 + 0.9701591241i\)
\(L(\frac12)\) \(\approx\) \(0.5487724692 + 0.9701591241i\)
\(L(1)\) \(\approx\) \(0.8085802720 + 0.5132681510i\)
\(L(1)\) \(\approx\) \(0.8085802720 + 0.5132681510i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.535 + 0.844i)T \)
3 \( 1 + (0.912 + 0.409i)T \)
5 \( 1 + (-0.358 - 0.933i)T \)
7 \( 1 + (0.130 + 0.991i)T \)
11 \( 1 + (-0.896 - 0.443i)T \)
13 \( 1 + (-0.885 + 0.465i)T \)
17 \( 1 + (0.931 + 0.363i)T \)
19 \( 1 + (0.545 + 0.837i)T \)
29 \( 1 + (0.0310 - 0.999i)T \)
31 \( 1 + (-0.191 + 0.981i)T \)
37 \( 1 + (0.999 + 0.0248i)T \)
41 \( 1 + (-0.0434 + 0.999i)T \)
43 \( 1 + (-0.449 - 0.893i)T \)
47 \( 1 + (0.460 + 0.887i)T \)
53 \( 1 + (0.980 - 0.197i)T \)
59 \( 1 + (0.275 + 0.961i)T \)
61 \( 1 + (0.0806 + 0.996i)T \)
67 \( 1 + (0.251 + 0.967i)T \)
71 \( 1 + (-0.982 + 0.185i)T \)
73 \( 1 + (0.0310 + 0.999i)T \)
79 \( 1 + (0.626 - 0.779i)T \)
83 \( 1 + (0.700 - 0.713i)T \)
89 \( 1 + (-0.215 + 0.976i)T \)
97 \( 1 + (-0.926 - 0.375i)T \)
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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

−23.17222174234527636883192153523, −22.23229527195324817380683801273, −21.28719329903141529287238135904, −20.30121317684848279653601478995, −19.96255588835537572058869421695, −19.07842272216295183217332432317, −18.27717304623073451713908425309, −17.75716041814197736376449003071, −16.58701113135018872946670264946, −15.39043785364322449914289575683, −14.48599304978845914505371781819, −13.67308118997595694253516785638, −12.879204686296607764498737152925, −11.95614696560958792615554259697, −10.88667978777158836108401707649, −10.10156834589706948286985431556, −9.47839714318856298174322754519, −8.016956069321380855225358774536, −7.541305958423888412735626658046, −6.97487289004817929432244646608, −4.9006584677319867870196168215, −3.71525252640069052503111259261, −2.95861927938163271247397374055, −2.16578743190923824229685352139, −0.66882444375661156782149668841, 1.41657594014358005694185964695, 2.6588556059489548355991339310, 4.12283134087748400824010520040, 5.12796186558841579003896397525, 5.75195997450004373695276983135, 7.43994255086645069570778838755, 8.08720645805207134662575662651, 8.704688617715746160407373209263, 9.57456956976757197879887594256, 10.24917386399396097863267091875, 11.73218550117754118577316077357, 12.76957209655240469434156610154, 13.73738592069140308765341534294, 14.70229114133384469414531421887, 15.28279247854836436621441407050, 16.25553306839003770632844511491, 16.539030387305185356113007583471, 17.888505024883721463886448610094, 18.91839740471928810141781087238, 19.27644945653003203975115458157, 20.341921093507105083143261139687, 21.17739136671300290995285913124, 21.93693367547352460677041976884, 23.26177292175086451868820347194, 24.07404560781749645663702969186

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