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Properties

Label 1-1441-1441.549-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.195 - 0.980i$
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  + (0.262 + 0.964i)2-s + (0.958 − 0.285i)3-s + (−0.861 + 0.506i)4-s + (−0.943 − 0.331i)5-s + (0.527 + 0.849i)6-s + (−0.926 + 0.377i)7-s + (−0.715 − 0.698i)8-s + (0.836 − 0.548i)9-s + (0.0724 − 0.997i)10-s + (−0.681 + 0.732i)12-s + (0.0724 + 0.997i)13-s + (−0.607 − 0.794i)14-s + (−0.998 − 0.0483i)15-s + (0.485 − 0.873i)16-s + (0.906 − 0.421i)17-s + (0.748 + 0.663i)18-s + ⋯
L(s)  = 1  + (0.262 + 0.964i)2-s + (0.958 − 0.285i)3-s + (−0.861 + 0.506i)4-s + (−0.943 − 0.331i)5-s + (0.527 + 0.849i)6-s + (−0.926 + 0.377i)7-s + (−0.715 − 0.698i)8-s + (0.836 − 0.548i)9-s + (0.0724 − 0.997i)10-s + (−0.681 + 0.732i)12-s + (0.0724 + 0.997i)13-s + (−0.607 − 0.794i)14-s + (−0.998 − 0.0483i)15-s + (0.485 − 0.873i)16-s + (0.906 − 0.421i)17-s + (0.748 + 0.663i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.195 - 0.980i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.195 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2076592801 - 0.1703009593i\)
\(L(\frac12)\) \(\approx\) \(0.2076592801 - 0.1703009593i\)
\(L(1)\) \(\approx\) \(0.9084806153 + 0.4567800063i\)
\(L(1)\) \(\approx\) \(0.9084806153 + 0.4567800063i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.262 + 0.964i)T \)
3 \( 1 + (0.958 - 0.285i)T \)
5 \( 1 + (-0.943 - 0.331i)T \)
7 \( 1 + (-0.926 + 0.377i)T \)
13 \( 1 + (0.0724 + 0.997i)T \)
17 \( 1 + (0.906 - 0.421i)T \)
19 \( 1 + (-0.120 + 0.992i)T \)
23 \( 1 + (-0.168 + 0.985i)T \)
29 \( 1 + (-0.836 - 0.548i)T \)
31 \( 1 + (-0.989 - 0.144i)T \)
37 \( 1 + (0.995 + 0.0965i)T \)
41 \( 1 + (-0.485 - 0.873i)T \)
43 \( 1 + (-0.0241 + 0.999i)T \)
47 \( 1 + (-0.354 + 0.935i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.981 + 0.192i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.0241 + 0.999i)T \)
71 \( 1 + (0.885 - 0.464i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.568 - 0.822i)T \)
83 \( 1 + (-0.215 + 0.976i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.644 + 0.764i)T \)
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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

−20.28422162551648936725746866692, −20.10183895120172901575374404919, −19.45938537573956976794912722176, −18.72647621773595199807318884657, −18.19692757478421106725272021368, −16.80009645257314659875112165792, −16.02424930043287042177185705024, −15.03117013733511139732019989045, −14.76111983643461109088505558514, −13.75133989623484328562135943115, −12.84382075895272480883382538182, −12.63268620142010407312112472662, −11.39077315319853758977541928030, −10.5600008905037733375925404014, −10.10290329801099563109997477733, −9.18479089056491460965827392381, −8.39663574152531259579151248353, −7.61933410844789809019397732564, −6.65732009757497720074845095261, −5.367303976163906147527086014321, −4.37100234291835122329561514219, −3.53659839125592424038389398397, −3.2024132904167957882781831670, −2.30086410269438572115444643262, −0.914624023604480015766319544394, 0.04909749124487705699624387069, 1.422668665716762928598466579265, 2.88955552113918721934451352098, 3.70674002287104590894972453215, 4.15299004874709467431991263189, 5.42209096448636864858775499882, 6.31992682528024290997793981479, 7.24609203247658069600078819831, 7.74925428101834485088077229583, 8.516632146703359407601774920588, 9.42398049892578046598768756107, 9.70135737949072471881939325458, 11.473484565405628376035263206838, 12.32802882458892431109152296873, 12.820200320314796215472575075198, 13.63748262961024322392143321953, 14.46632429774021587370838996855, 15.01330327772905574461597440221, 15.93914383461878308925711895927, 16.242094947699721002912448961522, 17.07812290268809218837898909524, 18.44374389019529695118421135713, 18.82686898089756068104966651389, 19.38441610818063604212446990774, 20.36620114680914819729525634308

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