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Properties

Label 2-2280-2280.1739-c0-0-1
Degree $2$
Conductor $2280$
Sign $0.479 - 0.877i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.479 - 0.877i$
Analytic conductor: \(1.13786\)
Root analytic conductor: \(1.06670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :0),\ 0.479 - 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.075760311\)
\(L(\frac12)\) \(\approx\) \(1.075760311\)
\(L(1)\) 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 \( 1 + (-0.342 - 0.939i)T + (-0.766 + 0.642i)T^{2} \)
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 \( 1 + (1.85 + 0.326i)T + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (-1.85 + 0.673i)T + (0.766 - 0.642i)T^{2} \)
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 \( 1 + (-0.642 + 1.11i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.272580960796440431289014409051, −8.574669254605632004379563649016, −8.094576019910047102200806130463, −7.33602156034883850545157319875, −6.09932249267384173965400438980, −5.49129435748860370801831110888, −4.67039949081327516220643547681, −3.68154581614383728359581236045, −2.36681037743054885810444304148, −1.34121011361084899678319738886, 1.06419783189964700216370013293, 2.30749605566216610043818485367, 2.93383514735622715000887810722, 3.62913989394797621275546760534, 5.17567725146148471006420647956, 6.49357198720502056771401483053, 6.91159832902721454151533176968, 7.59806707470813588638401233057, 8.287518673805953551301631698842, 9.147218906758228376117190687346

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