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Properties

Label 1-1441-1441.1019-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.946 + 0.323i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (−0.836 − 0.548i)2-s + (0.568 − 0.822i)3-s + (0.399 + 0.916i)4-s + (0.568 − 0.822i)5-s + (−0.926 + 0.377i)6-s + (−0.926 − 0.377i)7-s + (0.168 − 0.985i)8-s + (−0.354 − 0.935i)9-s + (−0.926 + 0.377i)10-s + (0.981 + 0.192i)12-s + (0.0724 − 0.997i)13-s + (0.568 + 0.822i)14-s + (−0.354 − 0.935i)15-s + (−0.681 + 0.732i)16-s + (−0.485 − 0.873i)17-s + (−0.215 + 0.976i)18-s + ⋯
L(s)  = 1  + (−0.836 − 0.548i)2-s + (0.568 − 0.822i)3-s + (0.399 + 0.916i)4-s + (0.568 − 0.822i)5-s + (−0.926 + 0.377i)6-s + (−0.926 − 0.377i)7-s + (0.168 − 0.985i)8-s + (−0.354 − 0.935i)9-s + (−0.926 + 0.377i)10-s + (0.981 + 0.192i)12-s + (0.0724 − 0.997i)13-s + (0.568 + 0.822i)14-s + (−0.354 − 0.935i)15-s + (−0.681 + 0.732i)16-s + (−0.485 − 0.873i)17-s + (−0.215 + 0.976i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.946 + 0.323i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.946 + 0.323i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07709548780 + 0.01280649124i\)
\(L(\frac12)\) \(\approx\) \(0.07709548780 + 0.01280649124i\)
\(L(1)\) \(\approx\) \(0.5092859572 - 0.4926794103i\)
\(L(1)\) \(\approx\) \(0.5092859572 - 0.4926794103i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.836 - 0.548i)T \)
3 \( 1 + (0.568 - 0.822i)T \)
5 \( 1 + (0.568 - 0.822i)T \)
7 \( 1 + (-0.926 - 0.377i)T \)
13 \( 1 + (0.0724 - 0.997i)T \)
17 \( 1 + (-0.485 - 0.873i)T \)
19 \( 1 + (-0.485 + 0.873i)T \)
23 \( 1 + (-0.989 - 0.144i)T \)
29 \( 1 + (-0.836 + 0.548i)T \)
31 \( 1 + (0.885 + 0.464i)T \)
37 \( 1 + (0.995 - 0.0965i)T \)
41 \( 1 + (-0.981 - 0.192i)T \)
43 \( 1 + (0.607 + 0.794i)T \)
47 \( 1 + (-0.262 + 0.964i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.906 - 0.421i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (-0.607 + 0.794i)T \)
71 \( 1 + (0.715 - 0.698i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.958 - 0.285i)T \)
83 \( 1 + (0.748 + 0.663i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.970 + 0.239i)T \)
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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

−20.31624904252315930971468012411, −19.48130510623850274558968985934, −19.06803079653609486205781669738, −18.34559871130540129503790518594, −17.32158864978191419508210184147, −16.751729732494692746854314132734, −15.91106150438747228612438855168, −15.21903052409219615346123113722, −14.80434503937010620237191636252, −13.78835750251268707171208168477, −13.30570515975454940178883759569, −11.7079908661687660781326748663, −10.9429738965489463582758384449, −10.15818369953787492443531249839, −9.61601853814924735379712305525, −9.01084658106230461690426791318, −8.217548582591808723496515931226, −7.149227105042016255122203221425, −6.336053387671512106546489876477, −5.82484662249100614695181958255, −4.571849786303895466149198557064, −3.559228660799281396240245439190, −2.40705444097456919992706066603, −1.97102476596617923339994719850, −0.02131765216371567203182162955, 0.791545626568499609441854122497, 1.6493560000424554421234856993, 2.60400792390434947599233098955, 3.33601947442304635751548313274, 4.379077138334667678268019591931, 5.902554685203805074952068400448, 6.53280514599866164797695873038, 7.57435834829256253759361107650, 8.17507452146464189824066663075, 8.99171585500886301689925584158, 9.67421016184607351249227319240, 10.2614524033330132789530525889, 11.40772402760855374504372088504, 12.52496090526799396019967063343, 12.677569364935692541468587656998, 13.46278248645991163004430102037, 14.207105022059503081875626613629, 15.55675164330554470840149277104, 16.238710675937887920142417318404, 17.03264185370274971647726733522, 17.728012367871267981741205768816, 18.35575420189041805401830326928, 19.1166004379956077781059851905, 19.9494787331961942043851622688, 20.32400336653069256130956354374

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