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Properties

Label 1-1441-1441.547-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.985 + 0.171i$
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  + (−0.989 + 0.144i)2-s + (0.644 + 0.764i)3-s + (0.958 − 0.285i)4-s + (−0.527 + 0.849i)5-s + (−0.748 − 0.663i)6-s + (0.861 + 0.506i)7-s + (−0.906 + 0.421i)8-s + (−0.168 + 0.985i)9-s + (0.399 − 0.916i)10-s + (0.836 + 0.548i)12-s + (−0.995 + 0.0965i)13-s + (−0.926 − 0.377i)14-s + (−0.989 + 0.144i)15-s + (0.836 − 0.548i)16-s + (−0.354 + 0.935i)17-s + (0.0241 − 0.999i)18-s + ⋯
L(s)  = 1  + (−0.989 + 0.144i)2-s + (0.644 + 0.764i)3-s + (0.958 − 0.285i)4-s + (−0.527 + 0.849i)5-s + (−0.748 − 0.663i)6-s + (0.861 + 0.506i)7-s + (−0.906 + 0.421i)8-s + (−0.168 + 0.985i)9-s + (0.399 − 0.916i)10-s + (0.836 + 0.548i)12-s + (−0.995 + 0.0965i)13-s + (−0.926 − 0.377i)14-s + (−0.989 + 0.144i)15-s + (0.836 − 0.548i)16-s + (−0.354 + 0.935i)17-s + (0.0241 − 0.999i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.985 + 0.171i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (547, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.985 + 0.171i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09620943618 + 1.113513520i\)
\(L(\frac12)\) \(\approx\) \(0.09620943618 + 1.113513520i\)
\(L(1)\) \(\approx\) \(0.6494022362 + 0.5530367332i\)
\(L(1)\) \(\approx\) \(0.6494022362 + 0.5530367332i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.989 + 0.144i)T \)
3 \( 1 + (0.644 + 0.764i)T \)
5 \( 1 + (-0.527 + 0.849i)T \)
7 \( 1 + (0.861 + 0.506i)T \)
13 \( 1 + (-0.995 + 0.0965i)T \)
17 \( 1 + (-0.354 + 0.935i)T \)
19 \( 1 + (0.836 + 0.548i)T \)
23 \( 1 + (0.906 + 0.421i)T \)
29 \( 1 + (0.715 + 0.698i)T \)
31 \( 1 + (0.906 - 0.421i)T \)
37 \( 1 + (0.607 - 0.794i)T \)
41 \( 1 + (0.354 + 0.935i)T \)
43 \( 1 + (-0.926 + 0.377i)T \)
47 \( 1 + (0.989 - 0.144i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.779 + 0.626i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.926 - 0.377i)T \)
71 \( 1 + (0.906 - 0.421i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.527 + 0.849i)T \)
83 \( 1 + (-0.607 - 0.794i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.861 + 0.506i)T \)
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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

−20.25346157086719419671284493588, −19.65097483863019266457089916967, −18.96144220313930533183957293815, −18.13972421158624803856749194665, −17.35283791045747691199798019833, −17.000375154216078377483454856072, −15.82219480433088511833711887364, −15.26065471597188711407523313749, −14.28668543967707274875566734754, −13.46515350927583734638293165751, −12.55189526834238787904717391289, −11.791752945361294519694619520227, −11.390692269717378597131225788861, −10.12533607759186907467943990002, −9.307257251208434230208859136490, −8.61264039409900869203985291516, −7.941938911189005465879801513299, −7.32408742759937746448908280726, −6.72976284013441157076270196079, −5.237093967083358212549732862497, −4.37269431172708305345197286114, −3.08788102606453096101099591160, −2.33394697576984705319337451134, −1.20297248783079196934245478892, −0.59344850808906889576659588427, 1.49775477951936766943903159680, 2.53196355135665513751605536625, 3.088226227530573482410829847400, 4.302215892863410609838260917687, 5.24996866871862185080622081267, 6.29505033901139547183668297956, 7.44764014274214328859478181806, 7.85969392390091026006958843284, 8.65062120056746081268756544704, 9.4589608805429362934911115664, 10.23057633147187507190446904334, 10.91477830529116060773440587802, 11.547756105202742509254955821533, 12.374677471202908332896017045564, 13.90737060667411991429539515697, 14.69797530929626317619207761415, 15.049045775394443450591845606372, 15.661160629475470471583944455834, 16.56215164511982479240897071360, 17.35542109528736064959831068754, 18.1639341334504507340077419608, 18.85943125835856989282423858268, 19.64012819487448913447181113080, 19.99417230202846389579573471215, 21.11567318927881239387234293753

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