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Properties

Label 1-1441-1441.729-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.217 + 0.976i$
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  + (0.715 + 0.698i)2-s + (−0.527 − 0.849i)3-s + (0.0241 + 0.999i)4-s + (0.926 + 0.377i)5-s + (0.215 − 0.976i)6-s + (−0.748 − 0.663i)7-s + (−0.681 + 0.732i)8-s + (−0.443 + 0.896i)9-s + (0.399 + 0.916i)10-s + (0.836 − 0.548i)12-s + (−0.748 − 0.663i)13-s + (−0.0724 − 0.997i)14-s + (−0.168 − 0.985i)15-s + (−0.998 + 0.0483i)16-s + (0.836 + 0.548i)17-s + (−0.943 + 0.331i)18-s + ⋯
L(s)  = 1  + (0.715 + 0.698i)2-s + (−0.527 − 0.849i)3-s + (0.0241 + 0.999i)4-s + (0.926 + 0.377i)5-s + (0.215 − 0.976i)6-s + (−0.748 − 0.663i)7-s + (−0.681 + 0.732i)8-s + (−0.443 + 0.896i)9-s + (0.399 + 0.916i)10-s + (0.836 − 0.548i)12-s + (−0.748 − 0.663i)13-s + (−0.0724 − 0.997i)14-s + (−0.168 − 0.985i)15-s + (−0.998 + 0.0483i)16-s + (0.836 + 0.548i)17-s + (−0.943 + 0.331i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.217 + 0.976i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.217 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.382409235 + 1.108293180i\)
\(L(\frac12)\) \(\approx\) \(1.382409235 + 1.108293180i\)
\(L(1)\) \(\approx\) \(1.242723384 + 0.4003506671i\)
\(L(1)\) \(\approx\) \(1.242723384 + 0.4003506671i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.715 + 0.698i)T \)
3 \( 1 + (-0.527 - 0.849i)T \)
5 \( 1 + (0.926 + 0.377i)T \)
7 \( 1 + (-0.748 - 0.663i)T \)
13 \( 1 + (-0.748 - 0.663i)T \)
17 \( 1 + (0.836 + 0.548i)T \)
19 \( 1 + (-0.262 - 0.964i)T \)
23 \( 1 + (-0.906 + 0.421i)T \)
29 \( 1 + (0.885 + 0.464i)T \)
31 \( 1 + (0.485 + 0.873i)T \)
37 \( 1 + (0.568 - 0.822i)T \)
41 \( 1 + (0.779 + 0.626i)T \)
43 \( 1 + (0.926 + 0.377i)T \)
47 \( 1 + (-0.443 + 0.896i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.836 + 0.548i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.926 - 0.377i)T \)
71 \( 1 + (0.981 - 0.192i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.0724 + 0.997i)T \)
83 \( 1 + (0.0241 - 0.999i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.995 + 0.0965i)T \)
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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

−20.88579765529180778672455704996, −20.108688156904213622126699494775, −19.10582019804576625218700611568, −18.46352353395756876140747518523, −17.53120529232780503250827686048, −16.56089990187741592351036699072, −16.12449023434995637637642612666, −15.14744751099826299855241301983, −14.37968305904395297601901636500, −13.79418228304220142677838501438, −12.65518572535024278180388172672, −12.17768275020024990160303733646, −11.57189538682607258781316212804, −10.36216766723487393981071876829, −9.768024963838926810533746070998, −9.55000517821163262654020155307, −8.44173550303023443990473450571, −6.68204815102886159600144839795, −6.02050382688066245982180563762, −5.46596390550809523779200319794, −4.64280079641663392797487709743, −3.83209502343229608857961017881, −2.763623536327982729312642160179, −2.054511553332721556797350604248, −0.62577413594688372105751851310, 1.06686807072477306880896229898, 2.493065822213220888488477292632, 3.03091219618332612751971686431, 4.34724233487573323195723611087, 5.34123335311675880720893490452, 6.013999499420718270659409880, 6.61406652245286519796028341658, 7.35096629710910959783120226478, 7.99102483275594497807880239793, 9.22341550932859811185577649368, 10.227912971098879789468394373806, 10.92735365102743452658743495352, 12.08014758857484500317349495569, 12.773970091548971096492288553943, 13.188820864714725933714610109867, 14.11300380922177768984533664091, 14.4476522818799930997012877060, 15.70528034326796257737538990513, 16.42630101325664319585079202848, 17.283198814869026159955623910728, 17.56571589966595056044460145263, 18.315972111366788300357772606413, 19.5056482957563353694212343566, 19.954052399041523131117312744986, 21.34975581030173944959384914273

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