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Properties

Label 1-1441-1441.642-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.0770 + 0.997i$
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  + (0.527 − 0.849i)2-s + (−0.906 − 0.421i)3-s + (−0.443 − 0.896i)4-s + (0.485 + 0.873i)5-s + (−0.836 + 0.548i)6-s + (−0.354 − 0.935i)7-s + (−0.995 − 0.0965i)8-s + (0.644 + 0.764i)9-s + (0.998 + 0.0483i)10-s + (0.0241 + 0.999i)12-s + (−0.354 − 0.935i)13-s + (−0.981 − 0.192i)14-s + (−0.0724 − 0.997i)15-s + (−0.607 + 0.794i)16-s + (−0.0241 + 0.999i)17-s + (0.989 − 0.144i)18-s + ⋯
L(s)  = 1  + (0.527 − 0.849i)2-s + (−0.906 − 0.421i)3-s + (−0.443 − 0.896i)4-s + (0.485 + 0.873i)5-s + (−0.836 + 0.548i)6-s + (−0.354 − 0.935i)7-s + (−0.995 − 0.0965i)8-s + (0.644 + 0.764i)9-s + (0.998 + 0.0483i)10-s + (0.0241 + 0.999i)12-s + (−0.354 − 0.935i)13-s + (−0.981 − 0.192i)14-s + (−0.0724 − 0.997i)15-s + (−0.607 + 0.794i)16-s + (−0.0241 + 0.999i)17-s + (0.989 − 0.144i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0770 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0770 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.0770 + 0.997i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (642, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.0770 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3248647977 - 0.3007426686i\)
\(L(\frac12)\) \(\approx\) \(-0.3248647977 - 0.3007426686i\)
\(L(1)\) \(\approx\) \(0.6814858281 - 0.5837499030i\)
\(L(1)\) \(\approx\) \(0.6814858281 - 0.5837499030i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.527 - 0.849i)T \)
3 \( 1 + (-0.906 - 0.421i)T \)
5 \( 1 + (0.485 + 0.873i)T \)
7 \( 1 + (-0.354 - 0.935i)T \)
13 \( 1 + (-0.354 - 0.935i)T \)
17 \( 1 + (-0.0241 + 0.999i)T \)
19 \( 1 + (0.943 - 0.331i)T \)
23 \( 1 + (-0.399 - 0.916i)T \)
29 \( 1 + (0.970 - 0.239i)T \)
31 \( 1 + (-0.215 - 0.976i)T \)
37 \( 1 + (-0.885 - 0.464i)T \)
41 \( 1 + (0.958 + 0.285i)T \)
43 \( 1 + (0.485 + 0.873i)T \)
47 \( 1 + (-0.644 - 0.764i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.0241 - 0.999i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.485 + 0.873i)T \)
71 \( 1 + (0.861 - 0.506i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.981 + 0.192i)T \)
83 \( 1 + (0.443 - 0.896i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.262 - 0.964i)T \)
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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

−21.322316773752923015045168387, −20.85556865179211510694755053875, −19.64850249611445900477466376238, −18.40190695301419731900384860313, −17.91913443317243317389848166241, −17.16522548999377424044830906398, −16.3579054689670903977806165370, −15.970908653256648022893557446321, −15.412111298547108561650710710104, −14.17359442498095598594960306926, −13.682343642737573767137877508629, −12.50402422526331327973500702865, −12.13827997453133708010126766996, −11.57267104099870292436213134137, −10.10105255105743559359886382406, −9.19647189625948556273410697043, −9.01678357216538759355775313422, −7.645496978459498602795849155801, −6.736067287368462827449256036144, −5.966038503454788612162163801622, −5.25821599530930008866016327375, −4.837056753270976493989829293709, −3.82602449742062398140487934719, −2.70216628529490124355884180697, −1.28882039435090173465340598615, 0.09427902768223589599326643387, 0.933443736545866424227933426782, 1.99439185666517299875256595094, 2.92130366125158309316116169291, 3.85264606501416323509418666645, 4.80231926237616317152184284261, 5.749654902543940772113243560676, 6.34819000511810359821458859978, 7.13056285588183931839604455387, 8.108541395783209871510291678086, 9.74514039953117549331974119748, 10.12304183965538382755237013642, 10.83112798240279570021780995122, 11.34686466358155959108047520613, 12.44723406400624130007663811654, 12.948848690913378001942903133200, 13.7139689225218186229924752874, 14.35630793783171702631614857861, 15.27808720211707357400446820248, 16.21622552621203458963911688327, 17.31522477108803348961679023423, 17.78081492477988470073407834450, 18.44665882696998525993895541841, 19.355247693188954755710340933792, 19.817889351807055965096875740831

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