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Properties

Label 1-1441-1441.328-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.421 + 0.906i$
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 + (0.0241 + 0.999i)3-s + (0.399 + 0.916i)4-s + (0.958 + 0.285i)5-s + (0.527 − 0.849i)6-s + (−0.970 + 0.239i)7-s + (0.168 − 0.985i)8-s + (−0.998 + 0.0483i)9-s + (−0.644 − 0.764i)10-s + (−0.906 + 0.421i)12-s + (−0.970 + 0.239i)13-s + (0.943 + 0.331i)14-s + (−0.262 + 0.964i)15-s + (−0.681 + 0.732i)16-s + (0.906 + 0.421i)17-s + (0.861 + 0.506i)18-s + ⋯
L(s)  = 1  + (−0.836 − 0.548i)2-s + (0.0241 + 0.999i)3-s + (0.399 + 0.916i)4-s + (0.958 + 0.285i)5-s + (0.527 − 0.849i)6-s + (−0.970 + 0.239i)7-s + (0.168 − 0.985i)8-s + (−0.998 + 0.0483i)9-s + (−0.644 − 0.764i)10-s + (−0.906 + 0.421i)12-s + (−0.970 + 0.239i)13-s + (0.943 + 0.331i)14-s + (−0.262 + 0.964i)15-s + (−0.681 + 0.732i)16-s + (0.906 + 0.421i)17-s + (0.861 + 0.506i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.421 + 0.906i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.421 + 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.008455093 + 0.6435852579i\)
\(L(\frac12)\) \(\approx\) \(1.008455093 + 0.6435852579i\)
\(L(1)\) \(\approx\) \(0.7206681822 + 0.1605479686i\)
\(L(1)\) \(\approx\) \(0.7206681822 + 0.1605479686i\)

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 \( 1 + (0.0241 + 0.999i)T \)
5 \( 1 + (0.958 + 0.285i)T \)
7 \( 1 + (-0.970 + 0.239i)T \)
13 \( 1 + (-0.970 + 0.239i)T \)
17 \( 1 + (0.906 + 0.421i)T \)
19 \( 1 + (-0.981 - 0.192i)T \)
23 \( 1 + (0.443 - 0.896i)T \)
29 \( 1 + (0.354 - 0.935i)T \)
31 \( 1 + (-0.715 + 0.698i)T \)
37 \( 1 + (0.748 - 0.663i)T \)
41 \( 1 + (0.485 - 0.873i)T \)
43 \( 1 + (0.958 + 0.285i)T \)
47 \( 1 + (0.998 - 0.0483i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.906 - 0.421i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.958 + 0.285i)T \)
71 \( 1 + (0.989 - 0.144i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.943 - 0.331i)T \)
83 \( 1 + (-0.399 + 0.916i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.0724 - 0.997i)T \)
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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

−20.055071734152674701532650463262, −19.48557447088115979866680783082, −18.80019988902093930103213331213, −18.142097065304904036354018137532, −17.298322477449164146323907171452, −16.87548369811512773260982419436, −16.26072872163734029135085988747, −15.00565376682230483549561868639, −14.36736317827045670875092526489, −13.569459577704118894957653444204, −12.82557629141839477150490552067, −12.150716638097232966876857502164, −10.97813846681695641091694147255, −10.11148868976566600177714242969, −9.43326010379979272605596609746, −8.827067189611521541803944197594, −7.67633370841195995168880898754, −7.21391960711019361115616045749, −6.24538334245245010961546936530, −5.81466438679659532641919201517, −4.870159343912989862010086403351, −3.06274690122191542317584613185, −2.29397488229408262702829110383, −1.30470551411848762823520840072, −0.48281611282545861082202523206, 0.61716566866251486219628131579, 2.22082491625368771334491901680, 2.683833348647526970472481435666, 3.61620592910881322994759968501, 4.58558067410339957592011972299, 5.79160028621359979728722209295, 6.46894450904787774317453993131, 7.49019265510515366378750305535, 8.661386254676122355468324979889, 9.31924490634001022464900198896, 9.802772894525976434389230304293, 10.51442183994882918607205105904, 11.01683761892096045850741564091, 12.37420986516320520923044776948, 12.66364798517575146093695634555, 13.915272374115287003504151209388, 14.68943155138819276909258750787, 15.54671513137004312774643174736, 16.41454013912656611837002465703, 17.01649861145569697100471639177, 17.43220208012134087685794574051, 18.57145273912254023165998508103, 19.21962250042261713868065347088, 19.82951212284984134489240127226, 20.833083278413034322754550292595

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