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Properties

Label 1-23e2-529.219-r0-0-0
Degree $1$
Conductor $529$
Sign $0.116 - 0.993i$
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  + (0.975 + 0.221i)2-s + (0.0806 − 0.996i)3-s + (0.901 + 0.432i)4-s + (−0.0929 − 0.995i)5-s + (0.299 − 0.954i)6-s + (0.524 − 0.851i)7-s + (0.783 + 0.621i)8-s + (−0.986 − 0.160i)9-s + (0.130 − 0.991i)10-s + (0.827 + 0.561i)11-s + (0.503 − 0.863i)12-s + (−0.404 − 0.914i)13-s + (0.700 − 0.713i)14-s + (−0.999 + 0.0124i)15-s + (0.626 + 0.779i)16-s + (−0.426 + 0.904i)17-s + ⋯
L(s)  = 1  + (0.975 + 0.221i)2-s + (0.0806 − 0.996i)3-s + (0.901 + 0.432i)4-s + (−0.0929 − 0.995i)5-s + (0.299 − 0.954i)6-s + (0.524 − 0.851i)7-s + (0.783 + 0.621i)8-s + (−0.986 − 0.160i)9-s + (0.130 − 0.991i)10-s + (0.827 + 0.561i)11-s + (0.503 − 0.863i)12-s + (−0.404 − 0.914i)13-s + (0.700 − 0.713i)14-s + (−0.999 + 0.0124i)15-s + (0.626 + 0.779i)16-s + (−0.426 + 0.904i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.116 - 0.993i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.116 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.014547534 - 1.792813126i\)
\(L(\frac12)\) \(\approx\) \(2.014547534 - 1.792813126i\)
\(L(1)\) \(\approx\) \(1.814487208 - 0.7924920986i\)
\(L(1)\) \(\approx\) \(1.814487208 - 0.7924920986i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.975 + 0.221i)T \)
3 \( 1 + (0.0806 - 0.996i)T \)
5 \( 1 + (-0.0929 - 0.995i)T \)
7 \( 1 + (0.524 - 0.851i)T \)
11 \( 1 + (0.827 + 0.561i)T \)
13 \( 1 + (-0.404 - 0.914i)T \)
17 \( 1 + (-0.426 + 0.904i)T \)
19 \( 1 + (0.606 - 0.794i)T \)
29 \( 1 + (-0.166 + 0.985i)T \)
31 \( 1 + (-0.215 - 0.976i)T \)
37 \( 1 + (0.179 + 0.983i)T \)
41 \( 1 + (-0.759 + 0.650i)T \)
43 \( 1 + (0.586 - 0.809i)T \)
47 \( 1 + (-0.0682 - 0.997i)T \)
53 \( 1 + (0.130 + 0.991i)T \)
59 \( 1 + (-0.998 - 0.0620i)T \)
61 \( 1 + (-0.191 + 0.981i)T \)
67 \( 1 + (-0.117 - 0.993i)T \)
71 \( 1 + (-0.535 + 0.844i)T \)
73 \( 1 + (-0.166 - 0.985i)T \)
79 \( 1 + (0.980 + 0.197i)T \)
83 \( 1 + (0.867 - 0.498i)T \)
89 \( 1 + (0.922 - 0.386i)T \)
97 \( 1 + (-0.907 + 0.421i)T \)
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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

−23.37839345032870341024674628409, −22.46868860044089058674732299877, −22.08566624346230360312880330109, −21.3682877949319183164366135659, −20.693447175267057151038888538473, −19.55219879235580067811581256856, −18.94546396208081122459103600306, −17.7673276499356564144352052770, −16.49740416818528918285486279245, −15.81243563344278681253492038233, −14.983434092881723400563520260772, −14.19862933138997836367356010440, −13.98696016136539124484386502215, −12.174634732317305848663637574220, −11.49510331886526841278091286728, −11.04020685520124639086367949304, −9.86527475791813658349665384879, −9.07351353173073477093974289283, −7.68150760841265317729186671055, −6.50074568102690802613886848099, −5.69696611739593960995600619050, −4.70176209175505611068584116872, −3.771271709485808899651212640275, −2.90469385256578558814516559061, −1.96699449981154079340565861256, 1.10073128823897967405025001628, 1.99214294536138982757029484432, 3.414836636969316510722794349944, 4.485115964872542269751689316069, 5.30252581252640732820812384311, 6.40002594831926804605682789699, 7.357980932825567414902891443946, 7.95687362558976375635418532413, 9.00081267304394605808954343066, 10.54638786830593907224665774173, 11.63585876836196811482344642604, 12.24962003148099104512041336179, 13.157297225659644556479319753527, 13.57005664301768946372991608553, 14.66763516239830081033046370155, 15.330931131140581089829266554504, 16.8008714574519777062971030389, 17.1166029582811956670529807893, 17.94249422968345360239730391856, 19.56490026637272864203208847513, 20.19626683500376167904909203271, 20.391593139144767776701211434444, 21.80677924453376587239916331966, 22.61270990034225621568595189525, 23.57372419099118147392948916267

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