↑

Properties

Label 1-1096-1096.475-r1-0-0
Degree $1$
Conductor $1096$
Sign $-0.791 + 0.611i$
Analytic cond. $117.781$
Root an. cond. $117.781$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.932 + 0.361i)3-s + (−0.850 − 0.526i)5-s + (0.982 + 0.183i)7-s + (0.739 − 0.673i)9-s + (0.445 − 0.895i)11-s + (−0.982 − 0.183i)13-s + (0.982 + 0.183i)15-s + (0.0922 − 0.995i)17-s + (−0.273 + 0.961i)19-s + (−0.982 + 0.183i)21-s + (−0.602 − 0.798i)23-s + (0.445 + 0.895i)25-s + (−0.445 + 0.895i)27-s + (−0.602 − 0.798i)29-s + (−0.273 − 0.961i)31-s + ⋯
L(s)  = 1  + (−0.932 + 0.361i)3-s + (−0.850 − 0.526i)5-s + (0.982 + 0.183i)7-s + (0.739 − 0.673i)9-s + (0.445 − 0.895i)11-s + (−0.982 − 0.183i)13-s + (0.982 + 0.183i)15-s + (0.0922 − 0.995i)17-s + (−0.273 + 0.961i)19-s + (−0.982 + 0.183i)21-s + (−0.602 − 0.798i)23-s + (0.445 + 0.895i)25-s + (−0.445 + 0.895i)27-s + (−0.602 − 0.798i)29-s + (−0.273 − 0.961i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $-0.791 + 0.611i$
Analytic conductor: \(117.781\)
Root analytic conductor: \(117.781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1096,\ (1:\ ),\ -0.791 + 0.611i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05222831933 - 0.1531017990i\)
\(L(\frac12)\) \(\approx\) \(-0.05222831933 - 0.1531017990i\)
\(L(1)\) \(\approx\) \(0.6409834181 - 0.1123174311i\)
\(L(1)\) \(\approx\) \(0.6409834181 - 0.1123174311i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
137 \( 1 \)
good3 \( 1 + (-0.932 + 0.361i)T \)
5 \( 1 + (-0.850 - 0.526i)T \)
7 \( 1 + (0.982 + 0.183i)T \)
11 \( 1 + (0.445 - 0.895i)T \)
13 \( 1 + (-0.982 - 0.183i)T \)
17 \( 1 + (0.0922 - 0.995i)T \)
19 \( 1 + (-0.273 + 0.961i)T \)
23 \( 1 + (-0.602 - 0.798i)T \)
29 \( 1 + (-0.602 - 0.798i)T \)
31 \( 1 + (-0.273 - 0.961i)T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + (0.273 - 0.961i)T \)
47 \( 1 + (0.739 - 0.673i)T \)
53 \( 1 + (-0.273 + 0.961i)T \)
59 \( 1 + (0.739 - 0.673i)T \)
61 \( 1 + (-0.739 - 0.673i)T \)
67 \( 1 + (0.982 + 0.183i)T \)
71 \( 1 + (0.445 + 0.895i)T \)
73 \( 1 + (-0.982 + 0.183i)T \)
79 \( 1 + (0.932 + 0.361i)T \)
83 \( 1 + (-0.0922 - 0.995i)T \)
89 \( 1 + (0.850 + 0.526i)T \)
97 \( 1 + (-0.445 + 0.895i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.983371593652382068819196949053, −21.05888675909493562306943831639, −19.7629328986805885831327830808, −19.545437922196936033267930715760, −18.45720142156491078046809409193, −17.65014159615049206666630787530, −17.31842220803618861182213956497, −16.35273910672902944964308878956, −15.33282287116108438890719094260, −14.808753488919985874786269615110, −13.95507494574011155886131353375, −12.71660175685004867782282402735, −12.11967055106434698486662444027, −11.46051933288118323820911365137, −10.75837561125127019968219824328, −10.04264116508756989549945721108, −8.7533827460310112151633774303, −7.64419065447303636449948041706, −7.23437385119807971202286667118, −6.452147337127443746758534357997, −5.16643693419243176148766214612, −4.59217542051845232556468085249, −3.684378202600674081081857100868, −2.16565841409352844684674963990, −1.36731754585283530490941894564, 0.051963780958385850782012170231, 0.78547160057232632549911911264, 2.06093017578532882567284542466, 3.6111213931032847300470825662, 4.333535845496551470018911512299, 5.160605002600441905586165223734, 5.79230468366555340199935272226, 7.01454425084658384680620530057, 7.8473987365888776436759719610, 8.64992109678896233229287962746, 9.61022388733561151699809960000, 10.58952015260404106014335239582, 11.401858970944835443434093284810, 11.9669795192408734019449004670, 12.43496939662498629341479385296, 13.7471780877472798729034993200, 14.674371432847970673242050594924, 15.38192654917762639640819195332, 16.19782626707256167751590226919, 16.92860912382059053213301469643, 17.33441265219184647455890288679, 18.67039735125601971016481292757, 18.84268800597255613224809515118, 20.37463171695500844085139152991, 20.57505242610428924201841895446

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