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Properties

Label 1-23e2-529.6-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  + (−0.709 + 0.704i)2-s + (0.481 − 0.876i)3-s + (0.00620 − 0.999i)4-s + (−0.896 + 0.443i)5-s + (0.275 + 0.961i)6-s + (0.912 + 0.409i)7-s + (0.700 + 0.713i)8-s + (−0.535 − 0.844i)9-s + (0.323 − 0.946i)10-s + (−0.492 − 0.870i)11-s + (−0.873 − 0.487i)12-s + (0.783 + 0.621i)13-s + (−0.935 + 0.352i)14-s + (−0.0434 + 0.999i)15-s + (−0.999 − 0.0124i)16-s + (−0.726 + 0.687i)17-s + ⋯
L(s)  = 1  + (−0.709 + 0.704i)2-s + (0.481 − 0.876i)3-s + (0.00620 − 0.999i)4-s + (−0.896 + 0.443i)5-s + (0.275 + 0.961i)6-s + (0.912 + 0.409i)7-s + (0.700 + 0.713i)8-s + (−0.535 − 0.844i)9-s + (0.323 − 0.946i)10-s + (−0.492 − 0.870i)11-s + (−0.873 − 0.487i)12-s + (0.783 + 0.621i)13-s + (−0.935 + 0.352i)14-s + (−0.0434 + 0.999i)15-s + (−0.999 − 0.0124i)16-s + (−0.726 + 0.687i)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: $\chi_{529} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.999 + 0.0118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.031910986 + 0.006128321526i\)
\(L(\frac12)\) \(\approx\) \(1.031910986 + 0.006128321526i\)
\(L(1)\) \(\approx\) \(0.8568292309 + 0.04466228060i\)
\(L(1)\) \(\approx\) \(0.8568292309 + 0.04466228060i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.709 + 0.704i)T \)
3 \( 1 + (0.481 - 0.876i)T \)
5 \( 1 + (-0.896 + 0.443i)T \)
7 \( 1 + (0.912 + 0.409i)T \)
11 \( 1 + (-0.492 - 0.870i)T \)
13 \( 1 + (0.783 + 0.621i)T \)
17 \( 1 + (-0.726 + 0.687i)T \)
19 \( 1 + (0.997 + 0.0744i)T \)
29 \( 1 + (0.980 + 0.197i)T \)
31 \( 1 + (0.999 - 0.0248i)T \)
37 \( 1 + (0.154 + 0.987i)T \)
41 \( 1 + (-0.616 - 0.787i)T \)
43 \( 1 + (-0.986 - 0.160i)T \)
47 \( 1 + (0.854 - 0.519i)T \)
53 \( 1 + (0.323 + 0.946i)T \)
59 \( 1 + (-0.215 - 0.976i)T \)
61 \( 1 + (-0.993 - 0.111i)T \)
67 \( 1 + (0.931 - 0.363i)T \)
71 \( 1 + (0.369 - 0.929i)T \)
73 \( 1 + (0.980 - 0.197i)T \)
79 \( 1 + (0.767 - 0.640i)T \)
83 \( 1 + (0.251 - 0.967i)T \)
89 \( 1 + (0.179 + 0.983i)T \)
97 \( 1 + (0.998 - 0.0496i)T \)
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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

−23.17800176890468725607201750405, −22.618474083081039635395489260411, −21.36099099305475224347925708213, −20.734649729143861013098564947944, −20.13476025036268707728794930210, −19.737797433857162078968648482099, −18.375448752803058016050678566800, −17.720065600458078600078093944604, −16.70577410528621608541831018094, −15.76770998010118077759876554820, −15.366940091909291830250743469567, −13.97967180342769700439391122891, −13.130246869959739514590856478577, −11.913165485949426269837000582650, −11.234606592687701297806683951879, −10.45674643340474141539125728141, −9.57983413333898874361928989209, −8.53303513674566039508536353396, −8.022543470814213596550593117052, −7.20033849247829298604043275224, −5.015881134288756608931101634394, −4.42779972380978352100249228287, −3.46397393129120455846889878599, −2.421247068101025562935328561682, −0.9809750087462463763548349562, 0.921450982953383586809756709308, 2.07679355160576046279222908261, 3.34393945303078640294910162442, 4.781617398154723813648873803006, 6.06995471784481675558582516398, 6.79152264610541639029607904922, 7.86131877431826175675992957364, 8.33395495016607130785060069820, 8.96413332897084645843219752367, 10.53111349798462566690273300158, 11.36099890677729312265121967802, 12.04704260339472391826291376308, 13.69761929812491157526362048662, 14.01694913217911954218349696223, 15.241449588986799943610755639275, 15.57816102644693902373561561927, 16.78630966172563619780704943775, 17.85131920220668758095469503081, 18.531244909049675108481836346785, 18.91590417403874505786601654828, 19.85470541946023386149894408935, 20.62761630392500693946303903999, 21.86517140913685809669792458706, 23.194164782779741267770122689680, 23.76722764495437437503067592577

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