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Properties

Label 2-1320-1320.269-c0-0-6
Degree $2$
Conductor $1320$
Sign $-0.0457 + 0.998i$
Analytic cond. $0.658765$
Root an. cond. $0.811643$
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  + (−0.951 + 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (−0.309 + 0.951i)6-s + (−1.11 − 1.53i)7-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (−0.809 + 0.587i)10-s + i·11-s − 0.999i·12-s + (1.53 + 1.11i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (0.587 + 0.809i)18-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (−0.309 + 0.951i)6-s + (−1.11 − 1.53i)7-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (−0.809 + 0.587i)10-s + i·11-s − 0.999i·12-s + (1.53 + 1.11i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (0.587 + 0.809i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.0457 + 0.998i$
Analytic conductor: \(0.658765\)
Root analytic conductor: \(0.811643\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :0),\ -0.0457 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8567990613\)
\(L(\frac12)\) \(\approx\) \(0.8567990613\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 - iT \)
good7 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
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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

−9.632909319029380346547072738431, −8.933733250593555479772305539131, −7.85202033026940464741421139291, −7.25369743180914858547779303887, −6.55791973036707589374137802570, −6.01014145108844656278539751027, −4.49740241408138204174585484321, −3.11580859631297778393023840917, −2.04287137544208671914709990149, −0.918139355861919354983507065002, 2.00966384209956548223347584112, 2.96291717974158269931619524793, 3.32963265763908458820165814671, 5.16756917408199311455554801516, 6.05357474136927135253688789543, 6.65526971913054950686315694261, 8.033719706954423348391075515288, 8.865856761268550627998754660374, 9.158756326746570048200231259223, 9.856703825076060224622179073392

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