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Properties

Label 1-4235-4235.467-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.933 + 0.357i$
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  + (0.647 − 0.761i)2-s + (0.207 − 0.978i)3-s + (−0.161 − 0.986i)4-s + (−0.610 − 0.791i)6-s + (−0.856 − 0.516i)8-s + (−0.913 − 0.406i)9-s + (−0.998 − 0.0475i)12-s + (−0.0570 − 0.998i)13-s + (−0.948 + 0.318i)16-s + (−0.846 + 0.532i)17-s + (−0.901 + 0.432i)18-s + (0.640 + 0.768i)19-s + (−0.814 − 0.580i)23-s + (−0.683 + 0.730i)24-s + (−0.797 − 0.603i)26-s + (−0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (0.647 − 0.761i)2-s + (0.207 − 0.978i)3-s + (−0.161 − 0.986i)4-s + (−0.610 − 0.791i)6-s + (−0.856 − 0.516i)8-s + (−0.913 − 0.406i)9-s + (−0.998 − 0.0475i)12-s + (−0.0570 − 0.998i)13-s + (−0.948 + 0.318i)16-s + (−0.846 + 0.532i)17-s + (−0.901 + 0.432i)18-s + (0.640 + 0.768i)19-s + (−0.814 − 0.580i)23-s + (−0.683 + 0.730i)24-s + (−0.797 − 0.603i)26-s + (−0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.933 + 0.357i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.933 + 0.357i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2365789254 + 0.04370756143i\)
\(L(\frac12)\) \(\approx\) \(0.2365789254 + 0.04370756143i\)
\(L(1)\) \(\approx\) \(0.7691019117 - 0.8259600633i\)
\(L(1)\) \(\approx\) \(0.7691019117 - 0.8259600633i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.647 - 0.761i)T \)
3 \( 1 + (0.207 - 0.978i)T \)
13 \( 1 + (-0.0570 - 0.998i)T \)
17 \( 1 + (-0.846 + 0.532i)T \)
19 \( 1 + (0.640 + 0.768i)T \)
23 \( 1 + (-0.814 - 0.580i)T \)
29 \( 1 + (-0.897 - 0.441i)T \)
31 \( 1 + (0.625 + 0.780i)T \)
37 \( 1 + (-0.703 - 0.710i)T \)
41 \( 1 + (0.0285 + 0.999i)T \)
43 \( 1 + (-0.755 + 0.654i)T \)
47 \( 1 + (0.901 + 0.432i)T \)
53 \( 1 + (-0.318 + 0.948i)T \)
59 \( 1 + (-0.851 - 0.524i)T \)
61 \( 1 + (0.179 + 0.983i)T \)
67 \( 1 + (0.690 + 0.723i)T \)
71 \( 1 + (0.696 - 0.717i)T \)
73 \( 1 + (-0.917 - 0.398i)T \)
79 \( 1 + (0.905 - 0.424i)T \)
83 \( 1 + (-0.491 - 0.870i)T \)
89 \( 1 + (-0.786 + 0.618i)T \)
97 \( 1 + (0.226 - 0.974i)T \)
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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

−18.164158611266688082003680716106, −17.264368859274616415081139203127, −16.91288596014960181531778399574, −16.06205140070963202963878322649, −15.58550353203192876491571785219, −15.16885460947145507889592683625, −14.174040956118692406048615652286, −13.83457750654980088937783055788, −13.25380670978681506064863788556, −12.09922167205693878848167831137, −11.56023510803094929998300066235, −10.982545392603854834262499832378, −9.84558609052932476422098574590, −9.25080199900645572179477128501, −8.706773958295729667872710328, −7.88842704065381859705793854901, −7.0480281039935117570165170341, −6.4417015374195638579224220378, −5.39748139143331158455169597403, −5.02409384820684520469077345452, −4.12122289481244286277814727600, −3.69564686391618055984470692894, −2.73477493774200434526900819728, −1.99402506479247993079821407132, −0.04939926204486853499795230799, 1.09049532838605144820520596108, 1.79574951895435677049284139147, 2.59571738980752620057726710355, 3.26336817215525957048138698601, 4.05547632921518824713520976463, 5.00480132116164553219831946803, 5.89649610909744277926991414558, 6.23560868190894851556898969357, 7.22750633738696481906307770202, 8.02221443742198401050833970027, 8.699549678296542457720955340251, 9.56848182415083971671773332151, 10.35406117424157858336841276418, 11.00706839027467270716805890650, 11.782061263102352466128101857520, 12.426643795485274749023285352781, 12.846108983818193832195730409695, 13.59957155712423936380188418620, 14.0982605266613955584505118709, 14.84094261783709735403289040418, 15.421438043292792251237049217331, 16.31179698745077803657204084414, 17.39379994285744488801889406149, 17.937112770179301505893111484428, 18.510988258933654123108596972162

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