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Properties

Label 2-1441-1441.130-c0-0-4
Degree $2$
Conductor $1441$
Sign $-0.740 + 0.672i$
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  + (1.03 − 0.749i)3-s + (−0.809 − 0.587i)4-s + (−0.613 − 1.88i)5-s + (0.303 + 0.220i)7-s + (0.193 − 0.594i)9-s + (0.0627 + 0.998i)11-s − 1.27·12-s + (0.598 − 1.84i)13-s + (−2.04 − 1.48i)15-s + (0.309 + 0.951i)16-s + (−0.613 + 1.88i)20-s + 0.477·21-s + (−2.37 + 1.72i)25-s + (0.147 + 0.454i)27-s + (−0.115 − 0.356i)28-s + ⋯
L(s)  = 1  + (1.03 − 0.749i)3-s + (−0.809 − 0.587i)4-s + (−0.613 − 1.88i)5-s + (0.303 + 0.220i)7-s + (0.193 − 0.594i)9-s + (0.0627 + 0.998i)11-s − 1.27·12-s + (0.598 − 1.84i)13-s + (−2.04 − 1.48i)15-s + (0.309 + 0.951i)16-s + (−0.613 + 1.88i)20-s + 0.477·21-s + (−2.37 + 1.72i)25-s + (0.147 + 0.454i)27-s + (−0.115 − 0.356i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.740 + 0.672i$
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: \((2,\ 1441,\ (\ :0),\ -0.740 + 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.150019013\)
\(L(\frac12)\) \(\approx\) \(1.150019013\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.0627 - 0.998i)T \)
131 \( 1 - T \)
good2 \( 1 + (0.809 + 0.587i)T^{2} \)
3 \( 1 + (-1.03 + 0.749i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.613 + 1.88i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.303 - 0.220i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (0.866 - 0.629i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.45T + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.688 - 0.500i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.613 + 1.88i)T + (-0.809 + 0.587i)T^{2} \)
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

−9.155786999369707790257738478183, −8.495721509357892481304645787715, −8.088865312986543604291567218916, −7.46982537733245148135627572525, −5.86969493210571586822513130117, −5.11231398550925489017114852985, −4.46145083118710187610710444179, −3.39566347412707845011821856633, −1.82389436877103448999361829499, −0.922909034634533040082382211927, 2.41893462899965295541316831119, 3.39927534570474310750864129164, 3.78632179803988028444268342243, 4.48162043615043325339080011998, 6.11641241900837189068104929944, 6.97503798496653352142998214815, 7.76882988503587348980797586272, 8.524862911230683839125787735407, 9.108750591278350623753618438957, 9.924204273812352068790782050361

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