Properties

Label 1-4235-4235.1319-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.959 - 0.282i$
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  + (0.580 + 0.814i)2-s + (−0.5 − 0.866i)3-s + (−0.327 + 0.945i)4-s + (0.415 − 0.909i)6-s + (−0.959 + 0.281i)8-s + (−0.5 + 0.866i)9-s + (0.981 − 0.189i)12-s + (0.654 + 0.755i)13-s + (−0.786 − 0.618i)16-s + (−0.0475 + 0.998i)17-s + (−0.995 + 0.0950i)18-s + (0.0475 + 0.998i)19-s + (0.786 + 0.618i)23-s + (0.723 + 0.690i)24-s + (−0.235 + 0.971i)26-s + 27-s + ⋯
L(s)  = 1  + (0.580 + 0.814i)2-s + (−0.5 − 0.866i)3-s + (−0.327 + 0.945i)4-s + (0.415 − 0.909i)6-s + (−0.959 + 0.281i)8-s + (−0.5 + 0.866i)9-s + (0.981 − 0.189i)12-s + (0.654 + 0.755i)13-s + (−0.786 − 0.618i)16-s + (−0.0475 + 0.998i)17-s + (−0.995 + 0.0950i)18-s + (0.0475 + 0.998i)19-s + (0.786 + 0.618i)23-s + (0.723 + 0.690i)24-s + (−0.235 + 0.971i)26-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.959 - 0.282i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.959 - 0.282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1329075383 + 0.9202767794i\)
\(L(\frac12)\) \(\approx\) \(-0.1329075383 + 0.9202767794i\)
\(L(1)\) \(\approx\) \(0.8757620576 + 0.4875731243i\)
\(L(1)\) \(\approx\) \(0.8757620576 + 0.4875731243i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.580 + 0.814i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (-0.0475 + 0.998i)T \)
19 \( 1 + (0.0475 + 0.998i)T \)
23 \( 1 + (0.786 + 0.618i)T \)
29 \( 1 + (-0.841 + 0.540i)T \)
31 \( 1 + (-0.981 - 0.189i)T \)
37 \( 1 + (0.327 + 0.945i)T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.995 - 0.0950i)T \)
53 \( 1 + (0.786 - 0.618i)T \)
59 \( 1 + (-0.580 + 0.814i)T \)
61 \( 1 + (-0.995 - 0.0950i)T \)
67 \( 1 + (0.995 - 0.0950i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (-0.928 + 0.371i)T \)
79 \( 1 + (-0.723 + 0.690i)T \)
83 \( 1 + (0.142 + 0.989i)T \)
89 \( 1 + (0.888 - 0.458i)T \)
97 \( 1 + (-0.959 + 0.281i)T \)
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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

−18.15764795813436908724436507645, −17.42000112380139620814529422650, −16.516424713120879056294055054588, −15.863768617723306643969768710296, −15.19520433973346604190105797999, −14.70816524482025525776904622303, −13.87966416508950263786286507056, −13.06994494569711399901984311444, −12.62216361122668351957460156265, −11.516167259684018529386403688714, −11.285725124901141610161513531446, −10.62600819753167333312458734816, −9.91398257722728722781343811865, −9.18942816090171695184306623909, −8.74961994826007477113209044106, −7.45927105076292720394102846614, −6.48164549936084596835195947565, −5.82352730889013736150623086264, −5.04550609467326058453853888650, −4.64377799039841733317654163957, −3.662612707069795185240339169820, −3.12854860966057359929565038452, −2.321272390213979697571338229311, −1.05722361414565411302946988009, −0.24604239992721801195377520089, 1.35728461177478504433325514218, 2.01888459181925904421107033474, 3.25479940222361467931192579384, 3.87230788000199825380014422896, 4.83546395989355016948045195494, 5.62343471046370348969362782686, 6.098686349516127210522837160811, 6.84410384047669957331085908362, 7.40086520499720417616983741617, 8.20823945000418905182698757422, 8.72018455345857883782302403900, 9.65786107973463847254834578702, 10.82498679419528498348370584312, 11.39843606678931191844158090199, 12.10683561155827405540330990623, 12.818181288666809309914360383795, 13.257150685311695617441551082534, 13.951073621381785116504750970242, 14.64582716214268190734350746603, 15.261827330374111804907999078298, 16.23450429972149375123478308178, 16.778042075757067570284523466166, 17.10370233805235250628910896321, 18.13150579886864646445856770603, 18.4531512337534817597792441735

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