Properties

Label 1-4235-4235.303-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.844 - 0.535i$
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  + (−0.836 + 0.548i)2-s + (0.207 − 0.978i)3-s + (0.398 − 0.917i)4-s + (0.362 + 0.931i)6-s + (0.170 + 0.985i)8-s + (−0.913 − 0.406i)9-s + (−0.814 − 0.580i)12-s + (−0.491 + 0.870i)13-s + (−0.683 − 0.730i)16-s + (−0.647 − 0.761i)17-s + (0.986 − 0.161i)18-s + (−0.851 + 0.524i)19-s + (−0.189 + 0.981i)23-s + (0.999 + 0.0380i)24-s + (−0.0665 − 0.997i)26-s + (−0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (−0.836 + 0.548i)2-s + (0.207 − 0.978i)3-s + (0.398 − 0.917i)4-s + (0.362 + 0.931i)6-s + (0.170 + 0.985i)8-s + (−0.913 − 0.406i)9-s + (−0.814 − 0.580i)12-s + (−0.491 + 0.870i)13-s + (−0.683 − 0.730i)16-s + (−0.647 − 0.761i)17-s + (0.986 − 0.161i)18-s + (−0.851 + 0.524i)19-s + (−0.189 + 0.981i)23-s + (0.999 + 0.0380i)24-s + (−0.0665 − 0.997i)26-s + (−0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.844 - 0.535i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.844 - 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7048062210 - 0.2047677535i\)
\(L(\frac12)\) \(\approx\) \(0.7048062210 - 0.2047677535i\)
\(L(1)\) \(\approx\) \(0.6402560227 - 0.05805029710i\)
\(L(1)\) \(\approx\) \(0.6402560227 - 0.05805029710i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.836 + 0.548i)T \)
3 \( 1 + (0.207 - 0.978i)T \)
13 \( 1 + (-0.491 + 0.870i)T \)
17 \( 1 + (-0.647 - 0.761i)T \)
19 \( 1 + (-0.851 + 0.524i)T \)
23 \( 1 + (-0.189 + 0.981i)T \)
29 \( 1 + (-0.564 + 0.825i)T \)
31 \( 1 + (-0.948 - 0.318i)T \)
37 \( 1 + (0.976 + 0.217i)T \)
41 \( 1 + (0.254 - 0.967i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 + (-0.986 - 0.161i)T \)
53 \( 1 + (-0.730 - 0.683i)T \)
59 \( 1 + (-0.964 - 0.263i)T \)
61 \( 1 + (-0.449 - 0.893i)T \)
67 \( 1 + (0.458 + 0.888i)T \)
71 \( 1 + (0.610 + 0.791i)T \)
73 \( 1 + (0.0190 + 0.999i)T \)
79 \( 1 + (0.272 - 0.962i)T \)
83 \( 1 + (0.996 - 0.0855i)T \)
89 \( 1 + (-0.723 - 0.690i)T \)
97 \( 1 + (-0.884 + 0.466i)T \)
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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

−18.34709736347805944223349752386, −17.75931838626939026677725526036, −16.99439531425876502446510056070, −16.648120800481472940853231792902, −15.78182712556996667846488484221, −15.13654811575381004839290596256, −14.684568587920596774497382953, −13.55308507155931522376152809691, −12.81255689320775830095938363800, −12.27042978650060052052439793377, −11.09579339867550642907355104503, −10.90840849264004557469984176692, −10.22969957070824007800677508061, −9.430121108102213198077852715378, −9.003260072678270261745072144191, −8.10581978344654161468466890252, −7.75394767765937562403464568802, −6.55120392223485155169483107298, −5.87925075812474428154257544875, −4.675606810099673245861862510617, −4.206831370458123461442276447290, −3.30384509993460561153225646877, −2.57612078541751793304026550621, −1.94802817427479534115407953158, −0.53375416369466152281175112981, 0.45171144798031771163315790749, 1.703327600012231933268144162651, 2.00562208145952817752030940584, 3.007325118913959762475869958739, 4.180481896512266805550498745, 5.21220331725013600373110068588, 5.92225935041568812178919492501, 6.68774395756836386398299062543, 7.19409526556092494638529534001, 7.81308253408212002635447378454, 8.54492716849100979520725985351, 9.31437283976689842535033489424, 9.64727207514665894265074919880, 10.969771884923647042976337554003, 11.27363453627452291083410714611, 12.14495203023505407823283933907, 12.907653902386969894488475338075, 13.70226243094662105014573635643, 14.413730919819756134741458505126, 14.77702454681122218663912755987, 15.75210407231666625577608555458, 16.42630859266414933664643349332, 17.12677389679410226803953344256, 17.62771233632585152621395327335, 18.38049386742158967126599332320

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