Properties

Label 1-1441-1441.16-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.861 + 0.507i$
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  + (−0.906 + 0.421i)2-s + (−0.861 + 0.506i)3-s + (0.644 − 0.764i)4-s + (0.995 + 0.0965i)5-s + (0.568 − 0.822i)6-s + (0.0241 − 0.999i)7-s + (−0.262 + 0.964i)8-s + (0.485 − 0.873i)9-s + (−0.943 + 0.331i)10-s + (−0.168 + 0.985i)12-s + (0.958 − 0.285i)13-s + (0.399 + 0.916i)14-s + (−0.906 + 0.421i)15-s + (−0.168 − 0.985i)16-s + (0.885 − 0.464i)17-s + (−0.0724 + 0.997i)18-s + ⋯
L(s)  = 1  + (−0.906 + 0.421i)2-s + (−0.861 + 0.506i)3-s + (0.644 − 0.764i)4-s + (0.995 + 0.0965i)5-s + (0.568 − 0.822i)6-s + (0.0241 − 0.999i)7-s + (−0.262 + 0.964i)8-s + (0.485 − 0.873i)9-s + (−0.943 + 0.331i)10-s + (−0.168 + 0.985i)12-s + (0.958 − 0.285i)13-s + (0.399 + 0.916i)14-s + (−0.906 + 0.421i)15-s + (−0.168 − 0.985i)16-s + (0.885 − 0.464i)17-s + (−0.0724 + 0.997i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.861 + 0.507i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.861 + 0.507i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9860155923 + 0.2684907289i\)
\(L(\frac12)\) \(\approx\) \(0.9860155923 + 0.2684907289i\)
\(L(1)\) \(\approx\) \(0.7296233610 + 0.1479553642i\)
\(L(1)\) \(\approx\) \(0.7296233610 + 0.1479553642i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.906 + 0.421i)T \)
3 \( 1 + (-0.861 + 0.506i)T \)
5 \( 1 + (0.995 + 0.0965i)T \)
7 \( 1 + (0.0241 - 0.999i)T \)
13 \( 1 + (0.958 - 0.285i)T \)
17 \( 1 + (0.885 - 0.464i)T \)
19 \( 1 + (-0.168 + 0.985i)T \)
23 \( 1 + (-0.262 - 0.964i)T \)
29 \( 1 + (-0.681 + 0.732i)T \)
31 \( 1 + (-0.262 + 0.964i)T \)
37 \( 1 + (0.926 + 0.377i)T \)
41 \( 1 + (0.885 + 0.464i)T \)
43 \( 1 + (0.399 - 0.916i)T \)
47 \( 1 + (-0.906 + 0.421i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.443 + 0.896i)T \)
61 \( 1 + T \)
67 \( 1 + (0.399 + 0.916i)T \)
71 \( 1 + (-0.262 + 0.964i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.995 + 0.0965i)T \)
83 \( 1 + (0.926 - 0.377i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.0241 - 0.999i)T \)
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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

−20.8950394974352291744834421246, −19.599686695470321250718121107814, −18.95728161197682295523018395860, −18.27817358448990530258818278555, −17.81153828240874606884963639215, −17.14451750460939164980264470206, −16.39390187762218651002154590049, −15.73371911824055516159528103491, −14.67892094552952921280404660909, −13.2732297444544622710623431872, −13.04862022918754256369780812376, −12.051685708067836587300675073179, −11.35296001123890536481752215541, −10.80065456931472726396007388835, −9.69004141287043671314389022120, −9.284675695017523520086783067797, −8.235082932223563886624595154829, −7.48778261851714596765451735312, −6.19973909324320890680554777127, −6.09542310793758354085455606055, −5.013296096020515455098628766588, −3.62162354834760261405721942782, −2.33238019082700903722156635867, −1.83319872925039643239411778544, −0.82634788562795317810402875269, 0.90717979168363284910193700842, 1.5027331737285368434460681059, 3.00033914346571424790647195574, 4.1524387327328750530025743143, 5.28914650514931378155292830904, 5.89419788693443413262776223365, 6.59458636215160071607207092197, 7.3778688171748184591805231337, 8.4238760795118285032756385657, 9.38145826257924585387571542450, 10.078535584377829300506875925, 10.5812499657456015267187156127, 11.11743798737161745276235726278, 12.24407299120513549285595076522, 13.19225016826003428412864955573, 14.317250985916651102011368461180, 14.6541327388826934304580089109, 15.97206768550137754117315791735, 16.48574272561866410111842902201, 16.89679096127099715968343470706, 17.80046754491105941077593538171, 18.22935077531741223600199231314, 18.93018904009722996652254787139, 20.33423676461295066771220552722, 20.62072165147592217009474436967

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