Properties

Label 1-23e2-529.13-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.892 + 0.451i$
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  + (0.813 + 0.581i)2-s + (0.998 + 0.0496i)3-s + (0.323 + 0.946i)4-s + (−0.709 + 0.704i)5-s + (0.783 + 0.621i)6-s + (−0.926 + 0.375i)7-s + (−0.287 + 0.957i)8-s + (0.995 + 0.0991i)9-s + (−0.986 + 0.160i)10-s + (−0.820 − 0.571i)11-s + (0.275 + 0.961i)12-s + (−0.556 + 0.831i)13-s + (−0.972 − 0.233i)14-s + (−0.743 + 0.668i)15-s + (−0.791 + 0.611i)16-s + (0.767 − 0.640i)17-s + ⋯
L(s)  = 1  + (0.813 + 0.581i)2-s + (0.998 + 0.0496i)3-s + (0.323 + 0.946i)4-s + (−0.709 + 0.704i)5-s + (0.783 + 0.621i)6-s + (−0.926 + 0.375i)7-s + (−0.287 + 0.957i)8-s + (0.995 + 0.0991i)9-s + (−0.986 + 0.160i)10-s + (−0.820 − 0.571i)11-s + (0.275 + 0.961i)12-s + (−0.556 + 0.831i)13-s + (−0.972 − 0.233i)14-s + (−0.743 + 0.668i)15-s + (−0.791 + 0.611i)16-s + (0.767 − 0.640i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.892 + 0.451i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.892 + 0.451i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4685569073 + 1.965347371i\)
\(L(\frac12)\) \(\approx\) \(0.4685569073 + 1.965347371i\)
\(L(1)\) \(\approx\) \(1.245976451 + 1.080685678i\)
\(L(1)\) \(\approx\) \(1.245976451 + 1.080685678i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.813 + 0.581i)T \)
3 \( 1 + (0.998 + 0.0496i)T \)
5 \( 1 + (-0.709 + 0.704i)T \)
7 \( 1 + (-0.926 + 0.375i)T \)
11 \( 1 + (-0.820 - 0.571i)T \)
13 \( 1 + (-0.556 + 0.831i)T \)
17 \( 1 + (0.767 - 0.640i)T \)
19 \( 1 + (-0.691 + 0.722i)T \)
29 \( 1 + (-0.449 + 0.893i)T \)
31 \( 1 + (0.251 + 0.967i)T \)
37 \( 1 + (0.931 + 0.363i)T \)
41 \( 1 + (0.606 - 0.794i)T \)
43 \( 1 + (0.645 + 0.763i)T \)
47 \( 1 + (-0.775 - 0.631i)T \)
53 \( 1 + (-0.986 - 0.160i)T \)
59 \( 1 + (0.867 + 0.498i)T \)
61 \( 1 + (-0.935 - 0.352i)T \)
67 \( 1 + (0.626 + 0.779i)T \)
71 \( 1 + (0.940 + 0.340i)T \)
73 \( 1 + (-0.449 - 0.893i)T \)
79 \( 1 + (0.664 - 0.747i)T \)
83 \( 1 + (0.798 + 0.601i)T \)
89 \( 1 + (-0.117 + 0.993i)T \)
97 \( 1 + (-0.873 + 0.487i)T \)
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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

−23.19385806624657605171475152691, −22.30938598376262746953344918179, −21.16601919201131993985648613119, −20.60948378103237846252710778511, −19.767807759180755296128564342883, −19.43102800918845683575237547985, −18.554506954160092930639361368282, −17.05508358970353741021334540492, −15.86053866783211505900101227355, −15.34234374256594463445200273398, −14.62532255203251505157482011779, −13.34252215682582813769252203698, −12.84971143042274117675928791797, −12.41552519995666782188413307641, −11.00194814465295873177375432231, −9.96888630851399753031203503270, −9.427559242812753715482558606168, −8.018664888782178387997829425129, −7.36564144028546445682509331159, −6.05046997040028177966828641984, −4.77240386830666938744864609141, −4.02133329377396212664792008259, −3.106739477445578029298238583179, −2.22257519709510791714040471992, −0.68606867242195906651922271518, 2.34887636736198719192116910711, 3.10377421180344626749332773287, 3.75716394981878600128171115770, 4.906797563687339730623312444353, 6.2190694652385664881741168154, 7.090613469997734689275249347836, 7.81697819802068839851031458594, 8.69680343081093668951873087030, 9.805225731104483739422703039750, 10.939285240006010915016529269700, 12.15811286999941156797453115599, 12.76884798814884888702009565400, 13.822147487887600735782291608252, 14.49937042041186255495677691792, 15.16251251266096962396867970608, 16.17981140324140253451877256621, 16.36513405859003467592791212585, 18.17264657072767511001193906071, 18.95460878638860206688209723066, 19.55466703388848223420689675240, 20.680130392602078203780629504402, 21.48158433896049374189582775031, 22.13879900936141302875541911770, 23.13855025066940239510067410136, 23.748768783156623504612724616533

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