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Properties

Label 1-23e2-529.50-r0-0-0
Degree $1$
Conductor $529$
Sign $0.565 - 0.824i$
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  + (0.545 − 0.837i)2-s + (−0.759 + 0.650i)3-s + (−0.404 − 0.914i)4-s + (−0.806 + 0.591i)5-s + (0.130 + 0.991i)6-s + (−0.999 + 0.0124i)7-s + (−0.986 − 0.160i)8-s + (0.154 − 0.987i)9-s + (0.0558 + 0.998i)10-s + (0.0310 + 0.999i)11-s + (0.901 + 0.432i)12-s + (0.586 − 0.809i)13-s + (−0.535 + 0.844i)14-s + (0.227 − 0.973i)15-s + (−0.673 + 0.739i)16-s + (−0.885 − 0.465i)17-s + ⋯
L(s)  = 1  + (0.545 − 0.837i)2-s + (−0.759 + 0.650i)3-s + (−0.404 − 0.914i)4-s + (−0.806 + 0.591i)5-s + (0.130 + 0.991i)6-s + (−0.999 + 0.0124i)7-s + (−0.986 − 0.160i)8-s + (0.154 − 0.987i)9-s + (0.0558 + 0.998i)10-s + (0.0310 + 0.999i)11-s + (0.901 + 0.432i)12-s + (0.586 − 0.809i)13-s + (−0.535 + 0.844i)14-s + (0.227 − 0.973i)15-s + (−0.673 + 0.739i)16-s + (−0.885 − 0.465i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.565 - 0.824i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.565 - 0.824i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7236539147 - 0.3811838896i\)
\(L(\frac12)\) \(\approx\) \(0.7236539147 - 0.3811838896i\)
\(L(1)\) \(\approx\) \(0.7672825750 - 0.2102807418i\)
\(L(1)\) \(\approx\) \(0.7672825750 - 0.2102807418i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.545 - 0.837i)T \)
3 \( 1 + (-0.759 + 0.650i)T \)
5 \( 1 + (-0.806 + 0.591i)T \)
7 \( 1 + (-0.999 + 0.0124i)T \)
11 \( 1 + (0.0310 + 0.999i)T \)
13 \( 1 + (0.586 - 0.809i)T \)
17 \( 1 + (-0.885 - 0.465i)T \)
19 \( 1 + (0.275 + 0.961i)T \)
29 \( 1 + (0.735 - 0.678i)T \)
31 \( 1 + (-0.0929 + 0.995i)T \)
37 \( 1 + (0.827 - 0.561i)T \)
41 \( 1 + (0.503 - 0.863i)T \)
43 \( 1 + (0.179 - 0.983i)T \)
47 \( 1 + (0.460 - 0.887i)T \)
53 \( 1 + (0.0558 - 0.998i)T \)
59 \( 1 + (0.912 + 0.409i)T \)
61 \( 1 + (-0.358 + 0.933i)T \)
67 \( 1 + (0.984 - 0.172i)T \)
71 \( 1 + (-0.239 - 0.970i)T \)
73 \( 1 + (0.735 + 0.678i)T \)
79 \( 1 + (-0.860 + 0.508i)T \)
83 \( 1 + (0.975 + 0.221i)T \)
89 \( 1 + (0.481 + 0.876i)T \)
97 \( 1 + (-0.982 - 0.185i)T \)
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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

−23.67402536192665655186447440820, −23.07637694191834534636932207950, −22.08782863523126493215046654803, −21.60966083526932133783700826008, −20.14981812474965880860499417599, −19.24325663125856818354726553052, −18.528337087554016647814240233256, −17.43426552009371582454402388932, −16.54640092414166350788919900530, −16.136588345764026365423162147149, −15.44058867548395159229731124677, −13.99211654787253515910560894921, −13.15295523494522362455917731359, −12.76175421496683928328292161569, −11.62124489865554204342108939224, −11.11094743931761809296546385441, −9.28599569011030522738596999960, −8.48277612223215631004360919804, −7.540024530033879889416231309799, −6.52169868908059112315944065716, −6.084578634999777467904048791795, −4.8361835932861047876034945056, −4.01538477800763133398544895614, −2.81746899398646139977156479006, −0.80756412206492079072838827029, 0.61202989812684354451527272451, 2.499029581058064269094882785069, 3.594434609022678610832902222993, 4.12638982525795853311505672693, 5.28776701485548070011618796786, 6.26883998986477614966827878656, 7.11283825192758335945285536217, 8.77178773513015725338865220063, 9.90441776810297106732437290632, 10.3882150479349218673131622569, 11.237338763256362241174560460046, 12.1646434144268257979141450241, 12.64577290365345118766501332725, 13.86187366367873848744616751990, 15.03346648845753902203790750486, 15.55443527537925961023299086689, 16.25305217676395678293160364976, 17.75854447608741718774391588102, 18.28995503570615187067338130586, 19.3669746526214145283087104276, 20.128814514280700903187515546804, 20.80355750658674175904066278884, 21.95622753966622245679413050657, 22.63564126252906113642700799460, 22.96577759111522416314230261539

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