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Properties

Label 2-2280-2280.389-c0-0-1
Degree $2$
Conductor $2280$
Sign $0.858 - 0.513i$
Analytic cond. $1.13786$
Root an. cond. $1.06670$
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  + (0.866 − 0.5i)2-s + (0.642 + 0.766i)3-s + (0.499 − 0.866i)4-s + (−0.342 + 0.939i)5-s + (0.939 + 0.342i)6-s − 0.999i·8-s + (−0.173 + 0.984i)9-s + (0.173 + 0.984i)10-s + (0.984 − 0.173i)12-s + (−0.939 + 0.342i)15-s + (−0.5 − 0.866i)16-s + (0.342 + 1.93i)17-s + (0.342 + 0.939i)18-s + (−0.766 + 0.642i)19-s + (0.642 + 0.766i)20-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.642 + 0.766i)3-s + (0.499 − 0.866i)4-s + (−0.342 + 0.939i)5-s + (0.939 + 0.342i)6-s − 0.999i·8-s + (−0.173 + 0.984i)9-s + (0.173 + 0.984i)10-s + (0.984 − 0.173i)12-s + (−0.939 + 0.342i)15-s + (−0.5 − 0.866i)16-s + (0.342 + 1.93i)17-s + (0.342 + 0.939i)18-s + (−0.766 + 0.642i)19-s + (0.642 + 0.766i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.858 - 0.513i$
Analytic conductor: \(1.13786\)
Root analytic conductor: \(1.06670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :0),\ 0.858 - 0.513i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.264583003\)
\(L(\frac12)\) \(\approx\) \(2.264583003\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.642 - 0.766i)T \)
5 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
good7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.342 - 1.93i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-1.62 + 0.592i)T + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.118 + 0.673i)T + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + (0.118 + 0.326i)T + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (-1.32 - 0.766i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.939 - 0.342i)T^{2} \)
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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

−9.543383247705591610513715096771, −8.453283488987490559486070256479, −7.83368478287904841471005413376, −6.71424609169294820471303720116, −6.09886198088306856272871139011, −5.09149427176472004958681891515, −4.10059906890221204191540797448, −3.65695570725468460830729023644, −2.77150435067731565496284490578, −1.88973792387138927930121641327, 1.21360836941754889974926416995, 2.67236847319660744738735534801, 3.27068178955310231131000996022, 4.52019819024399003805343361015, 5.02663586587485194682120008774, 6.02202160395056340625942208573, 7.06417157462235200641416286268, 7.34539730081024083982634123624, 8.253658601432532463636273500952, 8.943041436162457874351392036809

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