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Properties

Label 1-23e2-529.509-r0-0-0
Degree $1$
Conductor $529$
Sign $0.997 - 0.0722i$
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  + (−0.999 + 0.0124i)2-s + (−0.426 − 0.904i)3-s + (0.999 − 0.0248i)4-s + (−0.263 + 0.964i)5-s + (0.437 + 0.899i)6-s + (−0.117 − 0.993i)7-s + (−0.999 + 0.0372i)8-s + (−0.635 + 0.771i)9-s + (0.251 − 0.967i)10-s + (−0.471 + 0.882i)11-s + (−0.449 − 0.893i)12-s + (−0.896 − 0.443i)13-s + (0.130 + 0.991i)14-s + (0.984 − 0.172i)15-s + (0.998 − 0.0496i)16-s + (−0.993 + 0.111i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0124i)2-s + (−0.426 − 0.904i)3-s + (0.999 − 0.0248i)4-s + (−0.263 + 0.964i)5-s + (0.437 + 0.899i)6-s + (−0.117 − 0.993i)7-s + (−0.999 + 0.0372i)8-s + (−0.635 + 0.771i)9-s + (0.251 − 0.967i)10-s + (−0.471 + 0.882i)11-s + (−0.449 − 0.893i)12-s + (−0.896 − 0.443i)13-s + (0.130 + 0.991i)14-s + (0.984 − 0.172i)15-s + (0.998 − 0.0496i)16-s + (−0.993 + 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.997 - 0.0722i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.997 - 0.0722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5238158897 + 0.01895590223i\)
\(L(\frac12)\) \(\approx\) \(0.5238158897 + 0.01895590223i\)
\(L(1)\) \(\approx\) \(0.5226813504 - 0.06882590606i\)
\(L(1)\) \(\approx\) \(0.5226813504 - 0.06882590606i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.999 + 0.0124i)T \)
3 \( 1 + (-0.426 - 0.904i)T \)
5 \( 1 + (-0.263 + 0.964i)T \)
7 \( 1 + (-0.117 - 0.993i)T \)
11 \( 1 + (-0.471 + 0.882i)T \)
13 \( 1 + (-0.896 - 0.443i)T \)
17 \( 1 + (-0.993 + 0.111i)T \)
19 \( 1 + (0.955 - 0.293i)T \)
29 \( 1 + (0.700 - 0.713i)T \)
31 \( 1 + (0.995 + 0.0991i)T \)
37 \( 1 + (0.813 + 0.581i)T \)
41 \( 1 + (-0.885 + 0.465i)T \)
43 \( 1 + (0.798 - 0.601i)T \)
47 \( 1 + (-0.576 + 0.816i)T \)
53 \( 1 + (0.251 + 0.967i)T \)
59 \( 1 + (0.645 + 0.763i)T \)
61 \( 1 + (0.901 - 0.432i)T \)
67 \( 1 + (0.0806 + 0.996i)T \)
71 \( 1 + (0.0558 - 0.998i)T \)
73 \( 1 + (0.700 + 0.713i)T \)
79 \( 1 + (-0.935 + 0.352i)T \)
83 \( 1 + (0.524 - 0.851i)T \)
89 \( 1 + (0.751 + 0.659i)T \)
97 \( 1 + (0.980 + 0.197i)T \)
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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

−23.795649897135879700554318618543, −22.414163405880132576099701335, −21.54124895867850931657745962868, −21.03734156284230441509228806428, −20.04482917328181620551562376173, −19.38466033518038230654459552564, −18.29933540897813007690142413649, −17.51105338516695014156960991938, −16.59897983061609025159879837298, −15.97172430281487910084095888945, −15.54339629113079943902501287179, −14.42171755675218553862321084467, −12.86208378490418127415385516946, −11.79696463015487829929773253993, −11.51087190436613032079189369502, −10.25358104024118730436535693122, −9.40420575743882187104429686644, −8.80266286084313768309920721573, −8.04430448901798552913167805777, −6.60042644331130376354937164689, −5.569700538502185975649398940971, −4.84538519609317667291977007785, −3.38661677619629538623956262722, −2.27605731772882906500019774088, −0.60442861351727177636543219579, 0.77823191698105571968155239477, 2.21608458656516523356761542905, 2.957176790209111705444231116360, 4.67268697027705845709859502859, 6.151204555205887317690050850326, 7.00526913350788743848589122736, 7.433771776679706327394165129046, 8.19667728292318059701403124677, 9.82390399288577750859176638355, 10.36602547076846057136133740156, 11.28231549024613240803380180030, 11.985931297511340846355852130785, 13.071521446321591666773460097323, 14.0680010640501719538208981573, 15.14198324301105437505046915591, 15.94108042232241132234051457754, 17.17671256059894098807435120178, 17.632899888621530997654627756554, 18.24953609568643098400492404209, 19.25216745332378531153693953897, 19.81413438501604238746975454708, 20.49891649480906381386719319382, 22.001415230354771868407029173752, 22.78321745161185463112503503674, 23.5593504693914839349149980315

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