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Properties

Label 1-23e2-529.239-r0-0-0
Degree $1$
Conductor $529$
Sign $0.983 - 0.181i$
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  + (−0.0929 − 0.995i)2-s + (0.813 + 0.581i)3-s + (−0.982 + 0.185i)4-s + (0.346 + 0.938i)5-s + (0.503 − 0.863i)6-s + (0.995 − 0.0991i)7-s + (0.275 + 0.961i)8-s + (0.323 + 0.946i)9-s + (0.901 − 0.432i)10-s + (0.969 − 0.245i)11-s + (−0.907 − 0.421i)12-s + (0.299 − 0.954i)13-s + (−0.191 − 0.981i)14-s + (−0.263 + 0.964i)15-s + (0.931 − 0.363i)16-s + (−0.743 − 0.668i)17-s + ⋯
L(s)  = 1  + (−0.0929 − 0.995i)2-s + (0.813 + 0.581i)3-s + (−0.982 + 0.185i)4-s + (0.346 + 0.938i)5-s + (0.503 − 0.863i)6-s + (0.995 − 0.0991i)7-s + (0.275 + 0.961i)8-s + (0.323 + 0.946i)9-s + (0.901 − 0.432i)10-s + (0.969 − 0.245i)11-s + (−0.907 − 0.421i)12-s + (0.299 − 0.954i)13-s + (−0.191 − 0.981i)14-s + (−0.263 + 0.964i)15-s + (0.931 − 0.363i)16-s + (−0.743 − 0.668i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.983 - 0.181i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.983 - 0.181i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.940289652 - 0.1775261275i\)
\(L(\frac12)\) \(\approx\) \(1.940289652 - 0.1775261275i\)
\(L(1)\) \(\approx\) \(1.438643944 - 0.2006795454i\)
\(L(1)\) \(\approx\) \(1.438643944 - 0.2006795454i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.0929 - 0.995i)T \)
3 \( 1 + (0.813 + 0.581i)T \)
5 \( 1 + (0.346 + 0.938i)T \)
7 \( 1 + (0.995 - 0.0991i)T \)
11 \( 1 + (0.969 - 0.245i)T \)
13 \( 1 + (0.299 - 0.954i)T \)
17 \( 1 + (-0.743 - 0.668i)T \)
19 \( 1 + (-0.616 - 0.787i)T \)
29 \( 1 + (0.948 + 0.317i)T \)
31 \( 1 + (0.735 + 0.678i)T \)
37 \( 1 + (0.0558 + 0.998i)T \)
41 \( 1 + (-0.471 - 0.882i)T \)
43 \( 1 + (0.130 + 0.991i)T \)
47 \( 1 + (-0.775 - 0.631i)T \)
53 \( 1 + (0.901 + 0.432i)T \)
59 \( 1 + (-0.972 - 0.233i)T \)
61 \( 1 + (-0.977 + 0.209i)T \)
67 \( 1 + (0.179 - 0.983i)T \)
71 \( 1 + (-0.358 - 0.933i)T \)
73 \( 1 + (0.948 - 0.317i)T \)
79 \( 1 + (-0.426 + 0.904i)T \)
83 \( 1 + (-0.215 + 0.976i)T \)
89 \( 1 + (-0.635 + 0.771i)T \)
97 \( 1 + (0.0806 + 0.996i)T \)
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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

−23.87286323866757643955430227405, −23.09662359798463475282159588439, −21.63014050365996480187135187915, −21.1157413571832632247580073931, −20.01109252497770777965504948435, −19.23396993596712131312964463519, −18.30026082034745506688197495964, −17.4197173992150112583945643788, −16.94990565801694443527210393352, −15.781421244957212948688774280478, −14.823696654356400353278453456770, −14.19838269722620511161627267423, −13.50588935714150414892277684721, −12.58258465698629257392018329837, −11.70893622660226322293235605621, −10.033149622415978900462440105470, −8.99087630656704446239728277452, −8.5984862077117329312307994689, −7.81903987360220929709578370896, −6.65062960542414518543751501694, −5.95073097375981873985349228333, −4.47961749758734549584267864874, −4.066978074604621289118527848462, −1.96580547246457736796188831349, −1.24257550583858745432676223679, 1.407805702221451295844422109742, 2.52000085677236531451390812005, 3.22059636592007023369958315229, 4.30578874773657202460202834419, 5.13171470528087204304125940619, 6.70479077259558485787954741914, 8.00094762770963232558876076925, 8.72364610429857057097613571230, 9.603453050298863429131449908039, 10.63422921431089235887448413297, 10.98585613319337418817866470825, 12.03719813374508762849275316652, 13.538354321328938830487685612404, 13.84787194564124761781648582428, 14.77858098728814702998157581614, 15.43377099511777505751581987358, 16.99122957187810503456139326103, 17.809649789511107762531565101431, 18.461676361650503846098422668008, 19.602884429711068971531162229464, 19.99049252040490727341295053391, 21.05198000543159255259972762221, 21.580692830442205498517114199256, 22.312631439375537852348404586976, 23.05207603802339299017748939164

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