Properties

Label 1-1441-1441.285-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.764 + 0.644i$
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  + (−0.443 − 0.896i)2-s + (−0.0724 − 0.997i)3-s + (−0.607 + 0.794i)4-s + (0.644 + 0.764i)5-s + (−0.861 + 0.506i)6-s + (−0.995 + 0.0965i)7-s + (0.981 + 0.192i)8-s + (−0.989 + 0.144i)9-s + (0.399 − 0.916i)10-s + (0.836 + 0.548i)12-s + (−0.399 − 0.916i)13-s + (0.527 + 0.849i)14-s + (0.715 − 0.698i)15-s + (−0.262 − 0.964i)16-s + (0.779 + 0.626i)17-s + (0.568 + 0.822i)18-s + ⋯
L(s)  = 1  + (−0.443 − 0.896i)2-s + (−0.0724 − 0.997i)3-s + (−0.607 + 0.794i)4-s + (0.644 + 0.764i)5-s + (−0.861 + 0.506i)6-s + (−0.995 + 0.0965i)7-s + (0.981 + 0.192i)8-s + (−0.989 + 0.144i)9-s + (0.399 − 0.916i)10-s + (0.836 + 0.548i)12-s + (−0.399 − 0.916i)13-s + (0.527 + 0.849i)14-s + (0.715 − 0.698i)15-s + (−0.262 − 0.964i)16-s + (0.779 + 0.626i)17-s + (0.568 + 0.822i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.764 + 0.644i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.764 + 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4225976917 + 0.1543964599i\)
\(L(\frac12)\) \(\approx\) \(0.4225976917 + 0.1543964599i\)
\(L(1)\) \(\approx\) \(0.5916643046 - 0.2882123182i\)
\(L(1)\) \(\approx\) \(0.5916643046 - 0.2882123182i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.443 - 0.896i)T \)
3 \( 1 + (-0.0724 - 0.997i)T \)
5 \( 1 + (0.644 + 0.764i)T \)
7 \( 1 + (-0.995 + 0.0965i)T \)
13 \( 1 + (-0.399 - 0.916i)T \)
17 \( 1 + (0.779 + 0.626i)T \)
19 \( 1 + (-0.354 - 0.935i)T \)
23 \( 1 + (0.906 + 0.421i)T \)
29 \( 1 + (-0.989 - 0.144i)T \)
31 \( 1 + (0.681 + 0.732i)T \)
37 \( 1 + (-0.0241 + 0.999i)T \)
41 \( 1 + (0.262 - 0.964i)T \)
43 \( 1 + (-0.926 + 0.377i)T \)
47 \( 1 + (-0.885 - 0.464i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.998 - 0.0483i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.926 - 0.377i)T \)
71 \( 1 + (-0.120 - 0.992i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.970 - 0.239i)T \)
83 \( 1 + (-0.943 + 0.331i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.215 + 0.976i)T \)
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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

−20.6953386404089321263796389471, −19.86828360559351280007453224182, −19.042269275139177844093272508251, −18.33393723469616762820763873819, −17.061372362265392670579972062461, −16.82616321053406491177617079499, −16.329622150610654837469724190667, −15.60146046596894616921754760901, −14.62095913042648100306366975821, −14.107626161143313200815847474857, −13.1927434081697716530555450943, −12.360293797523301883378025990044, −11.19435377027433578397585753960, −10.12775526751194661919316939551, −9.65424068685932606725381341870, −9.1971201312921738990196175078, −8.390081319721554359742021302700, −7.31982811477056672633673814706, −6.250537729962712607328359565437, −5.76032789427358023368856466326, −4.821710221721045799762950809788, −4.19825979258165792558626370358, −3.010642662081060017834153583159, −1.61900652562456362955343491185, −0.218268622573170501924220867681, 1.118091910719240657607690252493, 2.0920365118539142183609813861, 3.05539373786535906314532879880, 3.27870520895146729421669749465, 5.028747269373594007730795882928, 5.94918269171545465628239126079, 6.8413991699886658698914167763, 7.48687700401669756694086313609, 8.44762160564281370378161978728, 9.33517211584181630524976491885, 10.10486636474276161472542479929, 10.76327410382249635631857474834, 11.61420100901477471371361880784, 12.493566343076646106170905501690, 13.11932112789349753814705530935, 13.51702521911698773365665173163, 14.55080335476098508315789503096, 15.421792282873387969758699149145, 16.90144598537208309986561136902, 17.161160748206217792164999624973, 18.04248541078339231107884300031, 18.60495191542838784440760116024, 19.43714176626458406771647851044, 19.58574477189803940744053859862, 20.72966209393275355182494945700

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