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Properties

Label 1-4235-4235.508-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.523 + 0.851i$
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  + (0.846 − 0.532i)2-s + (−0.207 − 0.978i)3-s + (0.432 − 0.901i)4-s + (−0.696 − 0.717i)6-s + (−0.113 − 0.993i)8-s + (−0.913 + 0.406i)9-s + (−0.971 − 0.235i)12-s + (0.336 − 0.941i)13-s + (−0.625 − 0.780i)16-s + (0.956 − 0.290i)17-s + (−0.556 + 0.830i)18-s + (−0.999 + 0.0190i)19-s + (−0.998 + 0.0475i)23-s + (−0.948 + 0.318i)24-s + (−0.217 − 0.976i)26-s + (0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (0.846 − 0.532i)2-s + (−0.207 − 0.978i)3-s + (0.432 − 0.901i)4-s + (−0.696 − 0.717i)6-s + (−0.113 − 0.993i)8-s + (−0.913 + 0.406i)9-s + (−0.971 − 0.235i)12-s + (0.336 − 0.941i)13-s + (−0.625 − 0.780i)16-s + (0.956 − 0.290i)17-s + (−0.556 + 0.830i)18-s + (−0.999 + 0.0190i)19-s + (−0.998 + 0.0475i)23-s + (−0.948 + 0.318i)24-s + (−0.217 − 0.976i)26-s + (0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.523 + 0.851i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.523 + 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8297120952 - 1.484398569i\)
\(L(\frac12)\) \(\approx\) \(-0.8297120952 - 1.484398569i\)
\(L(1)\) \(\approx\) \(0.8614040618 - 1.080812119i\)
\(L(1)\) \(\approx\) \(0.8614040618 - 1.080812119i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.846 - 0.532i)T \)
3 \( 1 + (-0.207 - 0.978i)T \)
13 \( 1 + (0.336 - 0.941i)T \)
17 \( 1 + (0.956 - 0.290i)T \)
19 \( 1 + (-0.999 + 0.0190i)T \)
23 \( 1 + (-0.998 + 0.0475i)T \)
29 \( 1 + (-0.921 - 0.389i)T \)
31 \( 1 + (0.380 - 0.924i)T \)
37 \( 1 + (0.475 - 0.879i)T \)
41 \( 1 + (0.985 + 0.170i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 + (0.556 + 0.830i)T \)
53 \( 1 + (-0.780 - 0.625i)T \)
59 \( 1 + (-0.640 - 0.768i)T \)
61 \( 1 + (-0.999 - 0.0380i)T \)
67 \( 1 + (0.618 + 0.786i)T \)
71 \( 1 + (0.0855 - 0.996i)T \)
73 \( 1 + (-0.986 - 0.161i)T \)
79 \( 1 + (0.00951 - 0.999i)T \)
83 \( 1 + (0.0570 + 0.998i)T \)
89 \( 1 + (-0.981 + 0.189i)T \)
97 \( 1 + (-0.980 - 0.198i)T \)
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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

−18.783757960270949919118910138910, −17.90321036156822551127697479784, −17.04386494710514619200048164187, −16.69791298354226803309011837606, −16.05617709896410143571536123401, −15.45472570234169603106230935204, −14.74016438401706083899030435025, −14.22234407007941659548722673722, −13.66932380226943787432634888143, −12.570762680796151261004898873258, −12.15268442725717966497972065659, −11.29642611394428520776359924754, −10.779659407912344316247235776433, −9.9262850411519860439416467188, −9.06606037546760808236248916170, −8.444502191719908132949654655259, −7.67019365662851769296895944614, −6.71945117829000482195479287518, −5.99823174990553558327218640633, −5.55819546138737762947982269565, −4.471510606101501026521490340997, −4.20687890104783541551217947584, −3.38176058637453126806152226981, −2.60361673211193448922547633021, −1.53883799474307069625907393797, 0.34154067702257249378308443185, 1.19736024748728314938442361769, 2.11720955577797216321271397111, 2.69567687077199676048082688197, 3.603802945322596193360977528526, 4.37544018318980843769462785268, 5.41581552710491981418068013271, 5.9527346199744170640431934960, 6.37751718654145816322200095238, 7.65444039005945288533428457414, 7.75961407093828074662228619152, 9.00498414939983208793902685568, 9.83864455560813209255074821436, 10.68698289472485894834905949330, 11.19183321053859452017288237343, 11.93366773650132252892338519058, 12.70543944778444915880269060460, 12.873321416576799970849102176350, 13.80532669592024216503649688391, 14.29692155154892585834379856984, 15.00214109698917489678167636630, 15.78492708812325063107394005832, 16.556677806606837344594230149798, 17.37006475570991543482727647906, 18.08532415938615982169348088965

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