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Properties

Label 1-4235-4235.1032-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.737 - 0.675i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.737 - 0.675i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1032, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.737 - 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4126018496 - 1.060992968i\)
\(L(\frac12)\) \(\approx\) \(0.4126018496 - 1.060992968i\)
\(L(1)\) \(\approx\) \(0.5926239826 - 0.4919911262i\)
\(L(1)\) \(\approx\) \(0.5926239826 - 0.4919911262i\)

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

−18.46339525719053379802525586446, −17.889511757310424511897057084115, −16.99766937290352578894206139357, −16.5179183441033726804210652707, −16.21338318359749989766691177861, −15.23092824170007012839367077819, −14.89685858000640177276667169501, −14.0168707995943589062957847620, −13.65448358065982614085627264596, −12.31397721219412810204720274199, −11.48580890075553670000281583888, −11.06706620948270460737239945198, −10.0434067789798083251641452892, −9.669416535180514448732236181376, −9.11529505684038567861674767542, −8.281687768207240596361880693500, −7.59467187553208694910193336726, −6.73204652611910432398283584473, −6.026230972418299232598101091589, −5.263614870091283682280003662738, −4.75821382900319679995766777993, −3.82844769973688758721987297136, −3.003425294703982833185829207468, −1.7780186597741432775614371053, −0.83037113836661975012538882072, 0.61088895068104097542242303810, 1.13346749194073849260839106960, 2.15490077085746430665600544336, 2.86070774734877057224882533413, 3.49085391512138043514569314125, 4.63655436314374490679043556997, 5.52225736714154110937483271998, 6.28130956903253937378206097484, 7.334648401092297195428719186033, 7.63799238610993091275007671801, 8.39572933259757255297691812997, 9.12208604080795326125289535146, 9.87258338221484346110701425178, 10.84874864791848621781141675762, 11.10631672826808336699493705751, 12.02160549050612996689729534206, 12.65762558728706891767120564223, 13.02335992725984273899416059000, 13.78018153332348362703045383454, 14.597992227093823755574551718409, 15.495499143909025368376019382422, 16.43730947468101354825282819174, 17.04037183335487873947013192746, 17.57346806865325395022669881939, 18.23681241116508631900175825065

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