Properties

Label 1-23e2-529.108-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.397 + 0.917i$
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  + (0.890 + 0.454i)2-s + (0.503 + 0.863i)3-s + (0.586 + 0.809i)4-s + (0.751 − 0.659i)5-s + (0.0558 + 0.998i)6-s + (0.227 + 0.973i)7-s + (0.154 + 0.987i)8-s + (−0.492 + 0.870i)9-s + (0.969 − 0.245i)10-s + (−0.977 − 0.209i)11-s + (−0.404 + 0.914i)12-s + (0.179 + 0.983i)13-s + (−0.239 + 0.970i)14-s + (0.948 + 0.317i)15-s + (−0.311 + 0.950i)16-s + (−0.449 − 0.893i)17-s + ⋯
L(s)  = 1  + (0.890 + 0.454i)2-s + (0.503 + 0.863i)3-s + (0.586 + 0.809i)4-s + (0.751 − 0.659i)5-s + (0.0558 + 0.998i)6-s + (0.227 + 0.973i)7-s + (0.154 + 0.987i)8-s + (−0.492 + 0.870i)9-s + (0.969 − 0.245i)10-s + (−0.977 − 0.209i)11-s + (−0.404 + 0.914i)12-s + (0.179 + 0.983i)13-s + (−0.239 + 0.970i)14-s + (0.948 + 0.317i)15-s + (−0.311 + 0.950i)16-s + (−0.449 − 0.893i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.397 + 0.917i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.397 + 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.634264039 + 2.487973133i\)
\(L(\frac12)\) \(\approx\) \(1.634264039 + 2.487973133i\)
\(L(1)\) \(\approx\) \(1.728298855 + 1.285046052i\)
\(L(1)\) \(\approx\) \(1.728298855 + 1.285046052i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.890 - 0.454i)T \)
3 \( 1 + (-0.503 - 0.863i)T \)
5 \( 1 + (-0.751 + 0.659i)T \)
7 \( 1 + (-0.227 - 0.973i)T \)
11 \( 1 + (0.977 + 0.209i)T \)
13 \( 1 + (-0.179 - 0.983i)T \)
17 \( 1 + (0.449 + 0.893i)T \)
19 \( 1 + (-0.323 + 0.946i)T \)
29 \( 1 + (-0.346 + 0.938i)T \)
31 \( 1 + (0.806 - 0.591i)T \)
37 \( 1 + (-0.0310 - 0.999i)T \)
41 \( 1 + (-0.901 + 0.432i)T \)
43 \( 1 + (-0.827 + 0.561i)T \)
47 \( 1 + (-0.203 + 0.979i)T \)
53 \( 1 + (-0.969 - 0.245i)T \)
59 \( 1 + (0.0434 + 0.999i)T \)
61 \( 1 + (0.287 + 0.957i)T \)
67 \( 1 + (-0.997 + 0.0744i)T \)
71 \( 1 + (-0.524 - 0.851i)T \)
73 \( 1 + (-0.346 - 0.938i)T \)
79 \( 1 + (-0.437 - 0.899i)T \)
83 \( 1 + (-0.545 - 0.837i)T \)
89 \( 1 + (0.616 + 0.787i)T \)
97 \( 1 + (-0.299 + 0.954i)T \)
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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

−23.11715103521141547723539939012, −22.62625122471572897293932077501, −21.380872585429479129151172449083, −20.71463742028204858173092769888, −20.02308937511674897404575134726, −19.198202487484260294866696648259, −18.14060700779285142696793456316, −17.72599426729084142831063272905, −16.31815642292641314249398471437, −15.00022973009905720533823823982, −14.519699919315936090572292206729, −13.628530968641702984003840830976, −13.06550824668951086863648384654, −12.40627666469563660357515689113, −10.87062598425101590115691875212, −10.58189548010021746279845856786, −9.47065306302799251296970068643, −7.88985377971146018879853395971, −7.24749616914119309117670171056, −6.17341285860316247225022639713, −5.47819291542254567867145265717, −3.97506752214457003589578712042, −3.04375339821949669708177795595, −2.15655164102488240266047865774, −1.15970467053703213408878328273, 2.208413404571396083015556814381, 2.69179705432039924083948121279, 4.12741726778305047001564867735, 5.11299847887277605174344571751, 5.431772033992633138219720781309, 6.72260214132524954906289634447, 8.07577491294839336402072372167, 8.85686162333283694566836083668, 9.5420440799837154677955313461, 10.91753933727601648942382986933, 11.7408549977676755847452594847, 12.84813453563399834149725449161, 13.73011844002853828733668909020, 14.186468792309990480718596989931, 15.47943642122719246708953681652, 15.77776620744504488069631265748, 16.63791736507277920995943527167, 17.59603567867501376426739586275, 18.6403734966716007218748483466, 20.01950166643256846382102607962, 20.72947258955422945880781964584, 21.511278185267905410281909662346, 21.707289070126145552798607069079, 22.74383726560862481785205253661, 23.943859454517346928269296128544

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