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Properties

Label 2-1441-1441.130-c0-0-2
Degree $2$
Conductor $1441$
Sign $0.994 - 0.108i$
Analytic cond. $0.719152$
Root an. cond. $0.848028$
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  + (−0.101 + 0.0738i)3-s + (−0.809 − 0.587i)4-s + (−0.263 − 0.809i)5-s + (1.60 + 1.16i)7-s + (−0.304 + 0.936i)9-s + (0.535 − 0.844i)11-s + 0.125·12-s + (−0.393 + 1.21i)13-s + (0.0865 + 0.0628i)15-s + (0.309 + 0.951i)16-s + (−0.263 + 0.809i)20-s − 0.249·21-s + (0.222 − 0.161i)25-s + (−0.0770 − 0.236i)27-s + (−0.613 − 1.88i)28-s + ⋯
L(s)  = 1  + (−0.101 + 0.0738i)3-s + (−0.809 − 0.587i)4-s + (−0.263 − 0.809i)5-s + (1.60 + 1.16i)7-s + (−0.304 + 0.936i)9-s + (0.535 − 0.844i)11-s + 0.125·12-s + (−0.393 + 1.21i)13-s + (0.0865 + 0.0628i)15-s + (0.309 + 0.951i)16-s + (−0.263 + 0.809i)20-s − 0.249·21-s + (0.222 − 0.161i)25-s + (−0.0770 − 0.236i)27-s + (−0.613 − 1.88i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.994 - 0.108i$
Analytic conductor: \(0.719152\)
Root analytic conductor: \(0.848028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :0),\ 0.994 - 0.108i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9915454837\)
\(L(\frac12)\) \(\approx\) \(0.9915454837\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.535 + 0.844i)T \)
131 \( 1 - T \)
good2 \( 1 + (0.809 + 0.587i)T^{2} \)
3 \( 1 + (0.101 - 0.0738i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.263 + 0.809i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-1.60 - 1.16i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.393 - 1.21i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-1.50 + 1.09i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.75T + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (1.17 + 0.856i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.263 + 0.809i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.190811396558022731939734285519, −9.103936171883904916611135227760, −8.310329069737734145397489889785, −7.68393675045939453450336042442, −6.07818434877950617814977123566, −5.45188911792079692250344237158, −4.69780851911102491214122188223, −4.26499465287914367144632120587, −2.35352747654180101931699450028, −1.33723571415909356380006815584, 1.06009824413208863222803088997, 2.82338287966799186893667330868, 3.86596972146155203400551304754, 4.46564662686727655071980929026, 5.38791945371876433586444349833, 6.68502879272659510890442279987, 7.65884067175106104011255754416, 7.72776061496307755923344039165, 8.865677225398879385161814528654, 9.704708442329950532809938594200

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