↑

Properties

Label 2-1680-420.419-c0-0-1
Degree $2$
Conductor $1680$
Sign $-i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + i·5-s + i·7-s − 9-s − 2·11-s + 15-s + 2i·17-s + 21-s − 25-s + i·27-s + 2i·33-s − 35-s − i·45-s − 49-s + 2·51-s + ⋯
L(s)  = 1  − i·3-s + i·5-s + i·7-s − 9-s − 2·11-s + 15-s + 2i·17-s + 21-s − 25-s + i·27-s + 2i·33-s − 35-s − i·45-s − 49-s + 2·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (1679, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6854605761\)
\(L(\frac12)\) \(\approx\) \(0.6854605761\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 - iT \)
7 \( 1 - iT \)
good11 \( 1 + 2T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 2iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 2T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.855918034035741112203991052252, −8.618568931385345382932258163986, −8.073814620998781890878017168443, −7.48555876178934797895645819617, −6.43545938638038371393113614245, −5.89403647237650208541669314541, −5.14328889679374909187033572304, −3.51672697259964223168166526173, −2.59016269178067947222452326918, −1.98010160627795497885338970204, 0.48849900682653517430019214264, 2.48241192453922548483147160691, 3.46597698526889206809255732170, 4.65555205882777062950951975978, 4.96223471342617099743442330928, 5.75044573659944102245542242543, 7.17153130126349435819151632051, 7.88148313102445387377003245981, 8.576735056484884037519338065336, 9.584838363646617594015196252237

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