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Properties

Label 1-1441-1441.1022-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.605 + 0.795i$
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  + (0.607 + 0.794i)2-s + (−0.989 − 0.144i)3-s + (−0.262 + 0.964i)4-s + (−0.168 − 0.985i)5-s + (−0.485 − 0.873i)6-s + (−0.981 − 0.192i)7-s + (−0.926 + 0.377i)8-s + (0.958 + 0.285i)9-s + (0.681 − 0.732i)10-s + (0.399 − 0.916i)12-s + (0.681 + 0.732i)13-s + (−0.443 − 0.896i)14-s + (0.0241 + 0.999i)15-s + (−0.861 − 0.506i)16-s + (−0.215 + 0.976i)17-s + (0.354 + 0.935i)18-s + ⋯
L(s)  = 1  + (0.607 + 0.794i)2-s + (−0.989 − 0.144i)3-s + (−0.262 + 0.964i)4-s + (−0.168 − 0.985i)5-s + (−0.485 − 0.873i)6-s + (−0.981 − 0.192i)7-s + (−0.926 + 0.377i)8-s + (0.958 + 0.285i)9-s + (0.681 − 0.732i)10-s + (0.399 − 0.916i)12-s + (0.681 + 0.732i)13-s + (−0.443 − 0.896i)14-s + (0.0241 + 0.999i)15-s + (−0.861 − 0.506i)16-s + (−0.215 + 0.976i)17-s + (0.354 + 0.935i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4423806884 + 0.8926752989i\)
\(L(\frac12)\) \(\approx\) \(0.4423806884 + 0.8926752989i\)
\(L(1)\) \(\approx\) \(0.7932188080 + 0.3015652933i\)
\(L(1)\) \(\approx\) \(0.7932188080 + 0.3015652933i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.607 + 0.794i)T \)
3 \( 1 + (-0.989 - 0.144i)T \)
5 \( 1 + (-0.168 - 0.985i)T \)
7 \( 1 + (-0.981 - 0.192i)T \)
13 \( 1 + (0.681 + 0.732i)T \)
17 \( 1 + (-0.215 + 0.976i)T \)
19 \( 1 + (0.748 + 0.663i)T \)
23 \( 1 + (0.644 - 0.764i)T \)
29 \( 1 + (-0.958 + 0.285i)T \)
31 \( 1 + (-0.0724 - 0.997i)T \)
37 \( 1 + (-0.998 + 0.0483i)T \)
41 \( 1 + (0.861 - 0.506i)T \)
43 \( 1 + (-0.715 - 0.698i)T \)
47 \( 1 + (0.568 - 0.822i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.995 - 0.0965i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.715 - 0.698i)T \)
71 \( 1 + (-0.970 - 0.239i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.885 + 0.464i)T \)
83 \( 1 + (-0.779 - 0.626i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.906 + 0.421i)T \)
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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

−20.34950074943744141245807332398, −19.45462474734141355445043010564, −18.826356504880358834687227359383, −18.10090106202248073290767182800, −17.626532755748530082611290090847, −16.19919608555631913460309179174, −15.65592802954782396567143832326, −15.120433418558061618407464582883, −13.96006794802774163030161538781, −13.24458488872720478622783557686, −12.60891730304168496319253395134, −11.57985946533693290363402153105, −11.2620722612305046784581693480, −10.42301975148310735790376660836, −9.783788266517401553666824412987, −9.043221084619733191170953757001, −7.2983348906477013124458242884, −6.75312478179670182081634803735, −5.812633424140933513663065479755, −5.31836874357415086897089702343, −4.14266063516460132808030284792, −3.27148254778141554378761716356, −2.73612531344617393296275378549, −1.26535032051916889828064070720, −0.27199301563318094172712311869, 0.70008118666989021945040881716, 1.97831173047261443044436426662, 3.76794239664392706074637777123, 3.98876887386505071214496931881, 5.20125468606105051073429624182, 5.72859965175224876000934139256, 6.56592876859300835770480059819, 7.186432627601201361627390835233, 8.23618914469777016089071407116, 9.037321864602003131110473599691, 9.90551346098990503323463841931, 11.04690217241346041350636230004, 11.91292003267762997806602796324, 12.56233674888069327037358475902, 13.118657360095470062859934273295, 13.70180664183527572237256722426, 14.92212757418616782963088919631, 15.85532433156095707618245590050, 16.25351656412571230827313417594, 16.895104204612274636907370529229, 17.32594379185805641932727905559, 18.50777613316396896956561112143, 19.04845122578863600598197980641, 20.30664537646379561561128925275, 20.96654277274056278753514724051

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