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Properties

Label 1-23e2-529.393-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.994 - 0.108i$
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  + (−0.743 − 0.668i)2-s + (0.767 − 0.640i)3-s + (0.105 + 0.994i)4-s + (−0.0434 − 0.999i)5-s + (−0.999 − 0.0372i)6-s + (0.626 − 0.779i)7-s + (0.586 − 0.809i)8-s + (0.179 − 0.983i)9-s + (−0.635 + 0.771i)10-s + (−0.616 − 0.787i)11-s + (0.717 + 0.696i)12-s + (0.392 + 0.919i)13-s + (−0.986 + 0.160i)14-s + (−0.673 − 0.739i)15-s + (−0.977 + 0.209i)16-s + (−0.952 − 0.305i)17-s + ⋯
L(s)  = 1  + (−0.743 − 0.668i)2-s + (0.767 − 0.640i)3-s + (0.105 + 0.994i)4-s + (−0.0434 − 0.999i)5-s + (−0.999 − 0.0372i)6-s + (0.626 − 0.779i)7-s + (0.586 − 0.809i)8-s + (0.179 − 0.983i)9-s + (−0.635 + 0.771i)10-s + (−0.616 − 0.787i)11-s + (0.717 + 0.696i)12-s + (0.392 + 0.919i)13-s + (−0.986 + 0.160i)14-s + (−0.673 − 0.739i)15-s + (−0.977 + 0.209i)16-s + (−0.952 − 0.305i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.994 - 0.108i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.994 - 0.108i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06212117970 - 1.143788272i\)
\(L(\frac12)\) \(\approx\) \(0.06212117970 - 1.143788272i\)
\(L(1)\) \(\approx\) \(0.6336268687 - 0.7190767003i\)
\(L(1)\) \(\approx\) \(0.6336268687 - 0.7190767003i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.743 - 0.668i)T \)
3 \( 1 + (0.767 - 0.640i)T \)
5 \( 1 + (-0.0434 - 0.999i)T \)
7 \( 1 + (0.626 - 0.779i)T \)
11 \( 1 + (-0.616 - 0.787i)T \)
13 \( 1 + (0.392 + 0.919i)T \)
17 \( 1 + (-0.952 - 0.305i)T \)
19 \( 1 + (0.299 - 0.954i)T \)
29 \( 1 + (-0.972 + 0.233i)T \)
31 \( 1 + (0.912 + 0.409i)T \)
37 \( 1 + (0.481 + 0.876i)T \)
41 \( 1 + (0.955 + 0.293i)T \)
43 \( 1 + (0.922 + 0.386i)T \)
47 \( 1 + (-0.990 + 0.136i)T \)
53 \( 1 + (-0.635 - 0.771i)T \)
59 \( 1 + (0.524 - 0.851i)T \)
61 \( 1 + (0.323 + 0.946i)T \)
67 \( 1 + (0.998 + 0.0496i)T \)
71 \( 1 + (0.154 + 0.987i)T \)
73 \( 1 + (-0.972 - 0.233i)T \)
79 \( 1 + (0.735 - 0.678i)T \)
83 \( 1 + (-0.926 - 0.375i)T \)
89 \( 1 + (0.0806 + 0.996i)T \)
97 \( 1 + (0.664 + 0.747i)T \)
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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

−24.22027121191015264658787832329, −22.896071761696681771230686017605, −22.427615726348174883556565317784, −21.18350057315780555305443554922, −20.47572925327964519955594873424, −19.559049371020022336578158800034, −18.64847192322171321787441766006, −18.087851295694044253052199349912, −17.28783879850883187992809906613, −15.810674335057550802497981665868, −15.47189589543578828518964262000, −14.77213109752343405326676409887, −14.13902696475944131194329569696, −12.946053549277639816210930819299, −11.32862654186978461139693679517, −10.62474728671386851148166438109, −9.88456626406630731996662482594, −8.98162844637093928343801440482, −7.92702203379617611565974946962, −7.58789671452483159447001996263, −6.1432341734335479465805995700, −5.2867413342727319557920492051, −4.09632399803536552851677821316, −2.63557388693131193799203430819, −1.94636410979982243027115388730, 0.716750418322298976302621620113, 1.589064233277605704513164287752, 2.66724006110962169339953638808, 3.884745578473630793959905414756, 4.75576727963829491637046845185, 6.55147962247051789323454346698, 7.54128362579789048742202477835, 8.312762973201475242038328881706, 8.926876566018547067502755697, 9.746644098054765437462270247386, 11.170619052654867143965197134190, 11.60276889895995334357488965830, 12.98747492464522212968856345592, 13.31969672631521951421872309744, 14.16787352609108554395368399336, 15.70495332294737381892350765571, 16.41704939421278081219462494531, 17.4666662356590058980780245656, 18.02803434191429952008410989404, 19.083506458283893921628293702786, 19.691159484248232842027469019482, 20.57353365788800216521649453087, 20.8994832212565807423063007149, 21.81399200381747478903141578301, 23.353147559484149592238226501

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