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Properties

Label 1-4235-4235.3084-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.882 + 0.470i$
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  + (0.710 + 0.703i)2-s + (0.104 − 0.994i)3-s + (0.00951 + 0.999i)4-s + (0.774 − 0.633i)6-s + (−0.696 + 0.717i)8-s + (−0.978 − 0.207i)9-s + (0.995 + 0.0950i)12-s + (0.736 − 0.676i)13-s + (−0.999 + 0.0190i)16-s + (−0.879 + 0.475i)17-s + (−0.548 − 0.836i)18-s + (−0.991 + 0.132i)19-s + (0.327 + 0.945i)23-s + (0.640 + 0.768i)24-s + (0.999 + 0.0380i)26-s + (−0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (0.710 + 0.703i)2-s + (0.104 − 0.994i)3-s + (0.00951 + 0.999i)4-s + (0.774 − 0.633i)6-s + (−0.696 + 0.717i)8-s + (−0.978 − 0.207i)9-s + (0.995 + 0.0950i)12-s + (0.736 − 0.676i)13-s + (−0.999 + 0.0190i)16-s + (−0.879 + 0.475i)17-s + (−0.548 − 0.836i)18-s + (−0.991 + 0.132i)19-s + (0.327 + 0.945i)23-s + (0.640 + 0.768i)24-s + (0.999 + 0.0380i)26-s + (−0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.882 + 0.470i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.882 + 0.470i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.253071113 + 0.5628986956i\)
\(L(\frac12)\) \(\approx\) \(2.253071113 + 0.5628986956i\)
\(L(1)\) \(\approx\) \(1.442705344 + 0.2360354976i\)
\(L(1)\) \(\approx\) \(1.442705344 + 0.2360354976i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.710 + 0.703i)T \)
3 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (0.736 - 0.676i)T \)
17 \( 1 + (-0.879 + 0.475i)T \)
19 \( 1 + (-0.991 + 0.132i)T \)
23 \( 1 + (0.327 + 0.945i)T \)
29 \( 1 + (0.941 - 0.336i)T \)
31 \( 1 + (-0.398 - 0.917i)T \)
37 \( 1 + (0.948 + 0.318i)T \)
41 \( 1 + (-0.362 - 0.931i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (-0.548 + 0.836i)T \)
53 \( 1 + (0.999 + 0.0190i)T \)
59 \( 1 + (0.988 - 0.151i)T \)
61 \( 1 + (0.964 + 0.263i)T \)
67 \( 1 + (-0.0475 + 0.998i)T \)
71 \( 1 + (-0.564 + 0.825i)T \)
73 \( 1 + (0.905 - 0.424i)T \)
79 \( 1 + (-0.0665 + 0.997i)T \)
83 \( 1 + (0.921 - 0.389i)T \)
89 \( 1 + (0.235 - 0.971i)T \)
97 \( 1 + (0.985 - 0.170i)T \)
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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

−18.30864238880431964889782115316, −17.855984994798470691406571288403, −16.637451296064538106315190429405, −16.21855744450975895725244131873, −15.50723861970759810928030634688, −14.74531846485838094331166140978, −14.394095813965290334691765972894, −13.50622516876933890220745999261, −13.015099328044407998444900241905, −12.03970599154515384394136924622, −11.384709665065752289855854951451, −10.80739081878055312225145219122, −10.33118161427997712989246316218, −9.42420825096735601309262798289, −8.88449629349869822429241338731, −8.25056186692229216112947646874, −6.6712200293392193918968871630, −6.42700702739029809223391379121, −5.34023749245911530409127183300, −4.64478921805646385997581274500, −4.21798792219628650363705884238, −3.39122127388839400092366748311, −2.63284279851684761897909679364, −1.92275605207944574651631781359, −0.67393712799392678754896493390, 0.733278675496003672746240645055, 1.984729349542275687868197665957, 2.60155192192132884887905067628, 3.57128406947370397599096599877, 4.18979602086800401806161301713, 5.285266944400382079678975177465, 5.926922709118223237391361860298, 6.480125090909409908136438222, 7.16829690206689603171209399245, 7.91565968925481055761890263960, 8.50825016450518137608116310998, 9.01431354070677795696645920177, 10.29723800647993062242980115681, 11.262338586800345713773516727171, 11.68283005581599044477934604808, 12.68377527336878457463020318674, 13.065522256980941328305984261706, 13.538832448680994548667729866039, 14.30213819728361902244175694725, 15.06506312851363636667670611365, 15.50081046612778461987243234546, 16.371530490396237694125508756133, 17.28755613635402505543860015443, 17.51186180691867839818552560021, 18.269408264129706803777668837313

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