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Properties

Label 1-23e2-529.311-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.213 - 0.976i$
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  + (0.645 − 0.763i)2-s + (−0.847 − 0.530i)3-s + (−0.166 − 0.985i)4-s + (−0.986 + 0.160i)5-s + (−0.952 + 0.305i)6-s + (0.392 + 0.919i)7-s + (−0.860 − 0.508i)8-s + (0.437 + 0.899i)9-s + (−0.514 + 0.857i)10-s + (0.975 − 0.221i)11-s + (−0.381 + 0.924i)12-s + (0.735 + 0.678i)13-s + (0.955 + 0.293i)14-s + (0.922 + 0.386i)15-s + (−0.944 + 0.329i)16-s + (0.0310 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.645 − 0.763i)2-s + (−0.847 − 0.530i)3-s + (−0.166 − 0.985i)4-s + (−0.986 + 0.160i)5-s + (−0.952 + 0.305i)6-s + (0.392 + 0.919i)7-s + (−0.860 − 0.508i)8-s + (0.437 + 0.899i)9-s + (−0.514 + 0.857i)10-s + (0.975 − 0.221i)11-s + (−0.381 + 0.924i)12-s + (0.735 + 0.678i)13-s + (0.955 + 0.293i)14-s + (0.922 + 0.386i)15-s + (−0.944 + 0.329i)16-s + (0.0310 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.213 - 0.976i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.213 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7871425905 - 0.9777293701i\)
\(L(\frac12)\) \(\approx\) \(0.7871425905 - 0.9777293701i\)
\(L(1)\) \(\approx\) \(0.8842639870 - 0.5746281619i\)
\(L(1)\) \(\approx\) \(0.8842639870 - 0.5746281619i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.645 - 0.763i)T \)
3 \( 1 + (-0.847 - 0.530i)T \)
5 \( 1 + (-0.986 + 0.160i)T \)
7 \( 1 + (0.392 + 0.919i)T \)
11 \( 1 + (0.975 - 0.221i)T \)
13 \( 1 + (0.735 + 0.678i)T \)
17 \( 1 + (0.0310 - 0.999i)T \)
19 \( 1 + (-0.426 - 0.904i)T \)
29 \( 1 + (0.606 + 0.794i)T \)
31 \( 1 + (0.783 + 0.621i)T \)
37 \( 1 + (0.867 - 0.498i)T \)
41 \( 1 + (-0.791 - 0.611i)T \)
43 \( 1 + (0.346 - 0.938i)T \)
47 \( 1 + (-0.576 + 0.816i)T \)
53 \( 1 + (-0.514 - 0.857i)T \)
59 \( 1 + (-0.404 - 0.914i)T \)
61 \( 1 + (0.992 + 0.123i)T \)
67 \( 1 + (-0.806 - 0.591i)T \)
71 \( 1 + (0.717 + 0.696i)T \)
73 \( 1 + (0.606 - 0.794i)T \)
79 \( 1 + (0.997 - 0.0744i)T \)
83 \( 1 + (-0.556 - 0.831i)T \)
89 \( 1 + (0.988 + 0.148i)T \)
97 \( 1 + (0.227 + 0.973i)T \)
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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

−23.47129027406227818208835307145, −23.046187273206616541522918668524, −22.42978076890581297091300032836, −21.32437191680362736765157828026, −20.62847008376069464780143362864, −19.737552854538190591503012267921, −18.380890391040663999346886082573, −17.31786129327551730532397398140, −16.87183324282013246239816877662, −16.12594302642886232993933650674, −15.14943651765446999002637256227, −14.72580165336657028362690149450, −13.441724931616153279642116508630, −12.4924403266603697577328552429, −11.717160487132194585019278656860, −10.98289186706942534734493902012, −9.90477201674391262814159460206, −8.44355421207502713018848423360, −7.83026473640932760859144328353, −6.63430789706625883150955261409, −6.01049864559142054333218593322, −4.668401733950782139305207304523, −4.109915458749040264155051369003, −3.47192586705847786035302769474, −1.05942525919024551139216175191, 0.80552992642075009754200710540, 1.98260115656941921901377834159, 3.1972858506561111674835749119, 4.42214567140921713691698602609, 5.09638300021451087248673984810, 6.32506658913068975725704188849, 6.91980560951044317469936701930, 8.423944153389561791907583552674, 9.29334539548851251110518033876, 10.80461419218950421914226095388, 11.39312852261908432764086272509, 11.908728469282080899549762116706, 12.5669442313126385188353243091, 13.72810427413114035221769993102, 14.52037901317111753966744972277, 15.62676365848168222821670210989, 16.21031289862770767538454555699, 17.620847932738980825565626614355, 18.44967505070336260281312923803, 19.085529952559963611363598631465, 19.709559501375111007153971792593, 20.866716744819974487909334917937, 21.84593519068659693016778004024, 22.33464387858144833414355779566, 23.221273750483364224745049696665

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