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Properties

Label 1-4235-4235.2984-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: $\chi_{4235} (2984, \cdot )$
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

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

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