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Properties

Label 1-1441-1441.380-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.852 + 0.522i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (−0.970 − 0.239i)2-s + (0.485 + 0.873i)3-s + (0.885 + 0.464i)4-s + (0.981 − 0.192i)5-s + (−0.262 − 0.964i)6-s + (−0.836 − 0.548i)7-s + (−0.748 − 0.663i)8-s + (−0.527 + 0.849i)9-s + (−0.998 − 0.0483i)10-s + (0.0241 + 0.999i)12-s + (−0.779 + 0.626i)13-s + (0.681 + 0.732i)14-s + (0.644 + 0.764i)15-s + (0.568 + 0.822i)16-s + (−0.943 − 0.331i)17-s + (0.715 − 0.698i)18-s + ⋯
L(s)  = 1  + (−0.970 − 0.239i)2-s + (0.485 + 0.873i)3-s + (0.885 + 0.464i)4-s + (0.981 − 0.192i)5-s + (−0.262 − 0.964i)6-s + (−0.836 − 0.548i)7-s + (−0.748 − 0.663i)8-s + (−0.527 + 0.849i)9-s + (−0.998 − 0.0483i)10-s + (0.0241 + 0.999i)12-s + (−0.779 + 0.626i)13-s + (0.681 + 0.732i)14-s + (0.644 + 0.764i)15-s + (0.568 + 0.822i)16-s + (−0.943 − 0.331i)17-s + (0.715 − 0.698i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.852 + 0.522i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (380, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.852 + 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.155764661 + 0.3257559292i\)
\(L(\frac12)\) \(\approx\) \(1.155764661 + 0.3257559292i\)
\(L(1)\) \(\approx\) \(0.8708217126 + 0.1303282263i\)
\(L(1)\) \(\approx\) \(0.8708217126 + 0.1303282263i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.970 - 0.239i)T \)
3 \( 1 + (0.485 + 0.873i)T \)
5 \( 1 + (0.981 - 0.192i)T \)
7 \( 1 + (-0.836 - 0.548i)T \)
13 \( 1 + (-0.779 + 0.626i)T \)
17 \( 1 + (-0.943 - 0.331i)T \)
19 \( 1 + (0.958 + 0.285i)T \)
23 \( 1 + (-0.399 - 0.916i)T \)
29 \( 1 + (0.644 - 0.764i)T \)
31 \( 1 + (0.861 - 0.506i)T \)
37 \( 1 + (0.989 - 0.144i)T \)
41 \( 1 + (0.607 + 0.794i)T \)
43 \( 1 + (-0.485 - 0.873i)T \)
47 \( 1 + (0.0724 + 0.997i)T \)
53 \( 1 + T \)
59 \( 1 + (0.568 + 0.822i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (-0.485 + 0.873i)T \)
71 \( 1 + (-0.215 - 0.976i)T \)
73 \( 1 + T \)
79 \( 1 + (0.485 + 0.873i)T \)
83 \( 1 + (0.715 + 0.698i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.998 - 0.0483i)T \)
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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

−20.197944307203284122896349594332, −19.7705622030923228052770743199, −19.15268213985113579869441762244, −18.16622800438408166795228408931, −17.8859640303543666455712797778, −17.23720739626948853245281989488, −16.20829930657647788704537283123, −15.37762457838034454148958839285, −14.70244800053220070189283744115, −13.79534846043131813051955444113, −13.076227415034940976590395504902, −12.26206493273187929902366421940, −11.46892093941041854335435418259, −10.28627084823577231755983348601, −9.67267427418842941946824351494, −9.040939760992255578886007256954, −8.297199034723554541574329988452, −7.28057499045274397033457276715, −6.69885315071932431328991218944, −5.97459328519092119119725108047, −5.23989490621325602054505848508, −3.221475676707380656755402327003, −2.60428305497318478844882136, −1.888434455403735967877568038728, −0.77282067117283618438972443878, 0.84622589964665553440582087796, 2.37526109332722623356933997281, 2.618167786305085446826191737982, 3.89107969527438191157011947277, 4.73394169569258716019355282288, 6.00652452003397355958869511176, 6.72248906719934705070165981170, 7.70153825024247380447818184664, 8.648923628273294876654218726805, 9.46287395893294649932264927689, 9.802440143322426187942386498031, 10.38176159164227415475007750540, 11.31591927098881192775406753891, 12.25641364588550728140671574336, 13.30579363898255351528365908369, 13.911826650071669598761994046653, 14.8368120771185628875237948971, 15.82084218550870706544660589742, 16.42283732169830558460464588068, 16.92078480528501312881701973483, 17.70024164837280229132251885722, 18.5625017174592893100337769501, 19.517524317604073976180649198793, 19.97243942822591453877653793581, 20.71670544943255951102421747807

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