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Properties

Label 2-1320-1320.677-c0-0-3
Degree $2$
Conductor $1320$
Sign $0.358 - 0.933i$
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  + (0.891 + 0.453i)2-s + (0.987 − 0.156i)3-s + (0.587 + 0.809i)4-s + (−0.987 − 0.156i)5-s + (0.951 + 0.309i)6-s + (−0.278 + 1.76i)7-s + (0.156 + 0.987i)8-s + (0.951 − 0.309i)9-s + (−0.809 − 0.587i)10-s + (−0.707 − 0.707i)11-s + (0.707 + 0.707i)12-s + (−1.04 + 1.44i)14-s − 15-s + (−0.309 + 0.951i)16-s + (0.987 + 0.156i)18-s + ⋯
L(s)  = 1  + (0.891 + 0.453i)2-s + (0.987 − 0.156i)3-s + (0.587 + 0.809i)4-s + (−0.987 − 0.156i)5-s + (0.951 + 0.309i)6-s + (−0.278 + 1.76i)7-s + (0.156 + 0.987i)8-s + (0.951 − 0.309i)9-s + (−0.809 − 0.587i)10-s + (−0.707 − 0.707i)11-s + (0.707 + 0.707i)12-s + (−1.04 + 1.44i)14-s − 15-s + (−0.309 + 0.951i)16-s + (0.987 + 0.156i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.358 - 0.933i$
Analytic conductor: \(0.658765\)
Root analytic conductor: \(0.811643\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :0),\ 0.358 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.033165268\)
\(L(\frac12)\) \(\approx\) \(2.033165268\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.891 - 0.453i)T \)
3 \( 1 + (-0.987 + 0.156i)T \)
5 \( 1 + (0.987 + 0.156i)T \)
11 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2} \)
13 \( 1 + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-1.59 + 1.16i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.550 + 0.280i)T + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.533 + 0.734i)T + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 - 0.587i)T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.0489i)T + (0.951 + 0.309i)T^{2} \)
79 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (1.44 - 0.734i)T + (0.587 - 0.809i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.142 + 0.278i)T + (-0.587 - 0.809i)T^{2} \)
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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

−9.726927373436168421202234501927, −8.796070871699569290721218367870, −8.194662540940183675838156934840, −7.79188845871960282496976806840, −6.62580022048941908089211677458, −5.82868485281887669280170349204, −4.87472274456652566373086517846, −3.87948826052191278685275118915, −2.93706340657894515702973591701, −2.35779895272830267225644039192, 1.36114031622218462077125930458, 2.94751633306370341042268659543, 3.50071019928916318536200897733, 4.43017004592833661638096596502, 4.86668839959856331350712980800, 6.71798338961160389364224004124, 7.17916347821391940178392775335, 7.80622563706974494253166485768, 8.888326821966228282683148077405, 10.10898670673558237360629152597

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