Properties

Label 1-4235-4235.794-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.994 + 0.107i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.290 + 0.956i)2-s + (−0.978 + 0.207i)3-s + (−0.830 − 0.556i)4-s + (0.0855 − 0.996i)6-s + (0.774 − 0.633i)8-s + (0.913 − 0.406i)9-s + (0.928 + 0.371i)12-s + (0.466 − 0.884i)13-s + (0.380 + 0.924i)16-s + (−0.749 + 0.662i)17-s + (0.123 + 0.992i)18-s + (−0.398 + 0.917i)19-s + (−0.235 + 0.971i)23-s + (−0.625 + 0.780i)24-s + (0.710 + 0.703i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (−0.290 + 0.956i)2-s + (−0.978 + 0.207i)3-s + (−0.830 − 0.556i)4-s + (0.0855 − 0.996i)6-s + (0.774 − 0.633i)8-s + (0.913 − 0.406i)9-s + (0.928 + 0.371i)12-s + (0.466 − 0.884i)13-s + (0.380 + 0.924i)16-s + (−0.749 + 0.662i)17-s + (0.123 + 0.992i)18-s + (−0.398 + 0.917i)19-s + (−0.235 + 0.971i)23-s + (−0.625 + 0.780i)24-s + (0.710 + 0.703i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.994 + 0.107i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (794, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.994 + 0.107i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03223330652 + 0.5975122043i\)
\(L(\frac12)\) \(\approx\) \(0.03223330652 + 0.5975122043i\)
\(L(1)\) \(\approx\) \(0.5145763860 + 0.3272781120i\)
\(L(1)\) \(\approx\) \(0.5145763860 + 0.3272781120i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.290 + 0.956i)T \)
3 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (0.466 - 0.884i)T \)
17 \( 1 + (-0.749 + 0.662i)T \)
19 \( 1 + (-0.398 + 0.917i)T \)
23 \( 1 + (-0.235 + 0.971i)T \)
29 \( 1 + (0.736 - 0.676i)T \)
31 \( 1 + (0.969 + 0.244i)T \)
37 \( 1 + (-0.345 + 0.938i)T \)
41 \( 1 + (0.516 + 0.856i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (0.123 - 0.992i)T \)
53 \( 1 + (-0.380 + 0.924i)T \)
59 \( 1 + (0.999 + 0.0190i)T \)
61 \( 1 + (-0.683 + 0.730i)T \)
67 \( 1 + (-0.981 - 0.189i)T \)
71 \( 1 + (-0.870 - 0.491i)T \)
73 \( 1 + (0.432 - 0.901i)T \)
79 \( 1 + (-0.548 + 0.836i)T \)
83 \( 1 + (-0.941 + 0.336i)T \)
89 \( 1 + (-0.580 - 0.814i)T \)
97 \( 1 + (-0.362 - 0.931i)T \)
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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

−17.82312916179693227698233179352, −17.74179360983456053581495157964, −16.90229510308686505314914180020, −16.10383348808403377844784357586, −15.710544277819800792059346491613, −14.3379659188809973216632451269, −13.81139761602876128397630852708, −13.01140230556704276874088912017, −12.49046125316862004739436591237, −11.75008293786173199297862054153, −11.21598493320016234559982619097, −10.68076785788977946125110720200, −10.01301143757059411224739256105, −9.060406565314841362745126665731, −8.671119444773714754981739110782, −7.554049700664599969226968596459, −6.85795702432812695894123141068, −6.17592398321331615360518294811, −5.11381483149389157275690493797, −4.48732382445154261200573418487, −3.976377478098806990077036925575, −2.69073477496437723714032805076, −2.10185885404767542711764763861, −1.1121237708838964236794280076, −0.29155884090561719028967163704, 0.92685176179821748911600502321, 1.668166505885078313896073843867, 3.18391209484173162942706448423, 4.21229067167607287930856800561, 4.61289754172464179336002738943, 5.717362648219078209913175244570, 5.98179052951722652225315856943, 6.66482946719761438783867934922, 7.561158536273546099629327655542, 8.20824607160495270054777162170, 8.91582833162419200280582664642, 9.98020332229190307125876677651, 10.2318512576073135709392144518, 11.02029169413063991326510172846, 11.85592901028132237832733331983, 12.69262202593030235201698853973, 13.290715900469924953551285798150, 13.98982606376142486979949434062, 15.02706058979581030555902750587, 15.464324584196213979836308008724, 15.94013509016161718675552162413, 16.81200668089847967305193977118, 17.23210337447147889131125897209, 17.93976460517217448711874782552, 18.26703768419289742397281135759

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