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Properties

Label 1-4235-4235.263-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.679 - 0.733i$
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  + (0.945 − 0.327i)2-s + (0.866 − 0.5i)3-s + (0.786 − 0.618i)4-s + (0.654 − 0.755i)6-s + (0.540 − 0.841i)8-s + (0.5 − 0.866i)9-s + (0.371 − 0.928i)12-s + (−0.989 + 0.142i)13-s + (0.235 − 0.971i)16-s + (−0.0950 − 0.995i)17-s + (0.189 − 0.981i)18-s + (−0.995 − 0.0950i)19-s + (0.971 + 0.235i)23-s + (0.0475 − 0.998i)24-s + (−0.888 + 0.458i)26-s − i·27-s + ⋯
L(s)  = 1  + (0.945 − 0.327i)2-s + (0.866 − 0.5i)3-s + (0.786 − 0.618i)4-s + (0.654 − 0.755i)6-s + (0.540 − 0.841i)8-s + (0.5 − 0.866i)9-s + (0.371 − 0.928i)12-s + (−0.989 + 0.142i)13-s + (0.235 − 0.971i)16-s + (−0.0950 − 0.995i)17-s + (0.189 − 0.981i)18-s + (−0.995 − 0.0950i)19-s + (0.971 + 0.235i)23-s + (0.0475 − 0.998i)24-s + (−0.888 + 0.458i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.679 - 0.733i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.679 - 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.704815462 - 3.900927031i\)
\(L(\frac12)\) \(\approx\) \(1.704815462 - 3.900927031i\)
\(L(1)\) \(\approx\) \(1.982457914 - 1.300440925i\)
\(L(1)\) \(\approx\) \(1.982457914 - 1.300440925i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.945 - 0.327i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.989 + 0.142i)T \)
17 \( 1 + (-0.0950 - 0.995i)T \)
19 \( 1 + (-0.995 - 0.0950i)T \)
23 \( 1 + (0.971 + 0.235i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (0.928 - 0.371i)T \)
37 \( 1 + (-0.618 + 0.786i)T \)
41 \( 1 + (0.654 - 0.755i)T \)
43 \( 1 + (0.540 - 0.841i)T \)
47 \( 1 + (-0.189 - 0.981i)T \)
53 \( 1 + (-0.971 + 0.235i)T \)
59 \( 1 + (0.327 - 0.945i)T \)
61 \( 1 + (-0.981 + 0.189i)T \)
67 \( 1 + (0.189 - 0.981i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (0.690 - 0.723i)T \)
79 \( 1 + (0.0475 + 0.998i)T \)
83 \( 1 + (-0.281 + 0.959i)T \)
89 \( 1 + (-0.580 - 0.814i)T \)
97 \( 1 + (0.540 - 0.841i)T \)
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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

−19.055107810406667161057497556507, −17.56147467063192349878152405611, −17.19770215163484252851640813385, −16.36200700980243050509474242985, −15.7155630939645301662104036534, −15.02778314593639663934840484051, −14.65279382735291376058586719999, −14.06890749845494174479810470374, −13.17531937542425317454509561412, −12.7569544517522543600617724439, −12.05623259717494506150856971311, −10.978925872918212422540927280698, −10.538863774735713195788344551658, −9.67052501437487689086824937379, −8.81058458825774661609351677641, −8.07691295359505884252826747454, −7.59113909013264810718307924387, −6.65926492500288329159178149303, −6.00326465867073642036298974882, −4.942121755128247025512007812325, −4.482582593789431392119866762226, −3.8233363433275906812288480075, −2.84943520218140069538983122262, −2.45481686196964482829483627687, −1.47511289390745530732199405382, 0.65278323720273112365945960362, 1.68091656579796316445048149041, 2.44637682052958964515309955807, 2.95563533256067914349258760067, 3.78145363299116717246628529366, 4.659982038595092008082076645157, 5.191884484907950932780886054351, 6.342490160245882562655655557952, 6.93797028571749088094865602973, 7.42588642815825734227149084274, 8.39854114786913937523403664280, 9.23050014151825251906511241854, 9.85963940114512846905978244758, 10.66802746300656144499172081120, 11.487981447741297840646618995722, 12.29283720291482500290119094444, 12.63049151861958792225090586749, 13.45609677397291097020232830130, 14.04286116740941288226466663409, 14.49337335715517232183907002441, 15.3872510904762772883567258525, 15.58613851525209320627754476836, 16.77475590565257395301675741182, 17.386422966171379986472483577396, 18.48178255321126487811849553992

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