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Properties

Label 1-1441-1441.659-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.884 + 0.467i$
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  + (−0.995 − 0.0965i)2-s + (0.779 + 0.626i)3-s + (0.981 + 0.192i)4-s + (−0.262 − 0.964i)5-s + (−0.715 − 0.698i)6-s + (0.989 − 0.144i)7-s + (−0.958 − 0.285i)8-s + (0.215 + 0.976i)9-s + (0.168 + 0.985i)10-s + (0.644 + 0.764i)12-s + (0.168 − 0.985i)13-s + (−0.998 + 0.0483i)14-s + (0.399 − 0.916i)15-s + (0.926 + 0.377i)16-s + (0.527 + 0.849i)17-s + (−0.120 − 0.992i)18-s + ⋯
L(s)  = 1  + (−0.995 − 0.0965i)2-s + (0.779 + 0.626i)3-s + (0.981 + 0.192i)4-s + (−0.262 − 0.964i)5-s + (−0.715 − 0.698i)6-s + (0.989 − 0.144i)7-s + (−0.958 − 0.285i)8-s + (0.215 + 0.976i)9-s + (0.168 + 0.985i)10-s + (0.644 + 0.764i)12-s + (0.168 − 0.985i)13-s + (−0.998 + 0.0483i)14-s + (0.399 − 0.916i)15-s + (0.926 + 0.377i)16-s + (0.527 + 0.849i)17-s + (−0.120 − 0.992i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.884 + 0.467i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.884 + 0.467i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1587556536 + 0.6398645459i\)
\(L(\frac12)\) \(\approx\) \(0.1587556536 + 0.6398645459i\)
\(L(1)\) \(\approx\) \(0.8509109004 + 0.09706987933i\)
\(L(1)\) \(\approx\) \(0.8509109004 + 0.09706987933i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.995 - 0.0965i)T \)
3 \( 1 + (0.779 + 0.626i)T \)
5 \( 1 + (-0.262 - 0.964i)T \)
7 \( 1 + (0.989 - 0.144i)T \)
13 \( 1 + (0.168 - 0.985i)T \)
17 \( 1 + (0.527 + 0.849i)T \)
19 \( 1 + (0.970 + 0.239i)T \)
23 \( 1 + (-0.607 + 0.794i)T \)
29 \( 1 + (-0.215 + 0.976i)T \)
31 \( 1 + (-0.943 + 0.331i)T \)
37 \( 1 + (-0.681 - 0.732i)T \)
41 \( 1 + (-0.926 + 0.377i)T \)
43 \( 1 + (-0.836 + 0.548i)T \)
47 \( 1 + (-0.748 - 0.663i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.0724 + 0.997i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.836 + 0.548i)T \)
71 \( 1 + (0.568 - 0.822i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.354 - 0.935i)T \)
83 \( 1 + (-0.485 - 0.873i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.443 - 0.896i)T \)
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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

−20.19885831271499820544680173113, −19.150336494173360955680917097879, −18.64292332645722529076985808803, −18.2461955339392446650311521024, −17.53999911663899042539486285592, −16.51724947129380980254100739017, −15.58704751416742393174453691259, −14.937493802135322960548506974618, −14.19354551350600340665274884721, −13.77101656582593811376224935808, −12.19272926280063102225740998596, −11.66644430497890748293484544960, −11.07044262469952178579686469401, −9.92369004459502736964935250514, −9.35107330860795292432481255224, −8.295114979126029373874840776, −7.869670000403009858271816065015, −7.02071456869922392337205028061, −6.55017439252415008223109077446, −5.34015853614678207447960684334, −3.89137189373217253096414913317, −2.93819996260904466478157291219, −2.13262268988192583413430514966, −1.42072536270398094297549621376, −0.14299638223360253070616163785, 1.35436717217827663976398224866, 1.72927585349460159888576385983, 3.220744789918013584814975193560, 3.787608198020808739467998460832, 5.08953512050992298521458251175, 5.61883341828435931686695970321, 7.2983981675877794822808957655, 7.9666688806341606208809335950, 8.35784631444590357138284824087, 9.14512740987870302075889934579, 9.946493856949587499263563106086, 10.64643174623019901888571968425, 11.45439115358801950511979539948, 12.33157256433216762855948821660, 13.16991712418988405271045796786, 14.244810662277386004612936582998, 15.008911156369319942995563148339, 15.692669559058529317579596178161, 16.36487648571556771134503130994, 17.04541662945877182161696172220, 17.85724314498471339251793173033, 18.57339771383916967697248482552, 19.68114625964592274318662676069, 20.1146718998969861268963578429, 20.524743162320770376424197341482

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