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Properties

Label 1-1441-1441.362-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.918 + 0.395i$
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  + (0.168 + 0.985i)2-s + (0.926 + 0.377i)3-s + (−0.943 + 0.331i)4-s + (−0.0724 + 0.997i)5-s + (−0.215 + 0.976i)6-s + (0.861 − 0.506i)7-s + (−0.485 − 0.873i)8-s + (0.715 + 0.698i)9-s + (−0.995 + 0.0965i)10-s + (−0.998 − 0.0483i)12-s + (−0.995 − 0.0965i)13-s + (0.644 + 0.764i)14-s + (−0.443 + 0.896i)15-s + (0.779 − 0.626i)16-s + (−0.836 − 0.548i)17-s + (−0.568 + 0.822i)18-s + ⋯
L(s)  = 1  + (0.168 + 0.985i)2-s + (0.926 + 0.377i)3-s + (−0.943 + 0.331i)4-s + (−0.0724 + 0.997i)5-s + (−0.215 + 0.976i)6-s + (0.861 − 0.506i)7-s + (−0.485 − 0.873i)8-s + (0.715 + 0.698i)9-s + (−0.995 + 0.0965i)10-s + (−0.998 − 0.0483i)12-s + (−0.995 − 0.0965i)13-s + (0.644 + 0.764i)14-s + (−0.443 + 0.896i)15-s + (0.779 − 0.626i)16-s + (−0.836 − 0.548i)17-s + (−0.568 + 0.822i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.918 + 0.395i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.918 + 0.395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.152528701 + 0.4437968488i\)
\(L(\frac12)\) \(\approx\) \(2.152528701 + 0.4437968488i\)
\(L(1)\) \(\approx\) \(1.122904116 + 0.7615722408i\)
\(L(1)\) \(\approx\) \(1.122904116 + 0.7615722408i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.168 + 0.985i)T \)
3 \( 1 + (0.926 + 0.377i)T \)
5 \( 1 + (-0.0724 + 0.997i)T \)
7 \( 1 + (0.861 - 0.506i)T \)
13 \( 1 + (-0.995 - 0.0965i)T \)
17 \( 1 + (-0.836 - 0.548i)T \)
19 \( 1 + (0.354 - 0.935i)T \)
23 \( 1 + (-0.681 - 0.732i)T \)
29 \( 1 + (-0.715 + 0.698i)T \)
31 \( 1 + (0.981 - 0.192i)T \)
37 \( 1 + (-0.607 - 0.794i)T \)
41 \( 1 + (-0.779 - 0.626i)T \)
43 \( 1 + (0.527 + 0.849i)T \)
47 \( 1 + (0.885 - 0.464i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.262 + 0.964i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (-0.527 + 0.849i)T \)
71 \( 1 + (0.120 - 0.992i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.970 - 0.239i)T \)
83 \( 1 + (-0.958 + 0.285i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.399 - 0.916i)T \)
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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

−20.3880619167686918548817598229, −19.94841227736145980257239368397, −19.1261044999795529513209454334, −18.54916342673391589401432556749, −17.54712235681128816995863711116, −17.131251509261198748004016431667, −15.60269285641525564432143783007, −15.07277103046581785493309035353, −14.1205008750917342944816685563, −13.62323517186594830918126303242, −12.75081772745366217721785578895, −12.05907240250612767698648936569, −11.68448221775727609396906066996, −10.33873702987527420053456199135, −9.57118986062819903780961602276, −8.90073651325797676981421137933, −8.19531645542229776298193214155, −7.641014444943608186700849035747, −6.076081395729330698219135563341, −5.11498883748868716143401490470, −4.368525981826086351479906326403, −3.616624022327047613184801762610, −2.358730687501827222005510175074, −1.848171099560932449027357130984, −1.017333359814027754374345034073, 0.355477644272487807659644072164, 2.10356265226405563160618293890, 2.918914484280205244653647557880, 3.96614478508999621023198877516, 4.60319312319957924391803261609, 5.424352341615456773871062849, 6.89087249554514365374861192039, 7.169546548812534292276126679479, 7.9741016758716967291836170762, 8.75149600844827205406914972095, 9.63621426539103513449715823488, 10.36972743257072353666142599940, 11.23365593410226941459727890187, 12.352035686259069848105028253910, 13.54242999850556673998052256031, 13.92009048330495971822466395680, 14.628619645844121667298022609862, 15.13600764892400468954403627152, 15.79493176487494782859494901411, 16.70001040365711943173853773322, 17.6934713549945334498303185142, 18.08943632659638630728075394526, 19.05196355572558611163136408453, 19.789157308505351130771711968158, 20.62013014291595524218763963181

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