↑

Properties

Label 2-1664-104.5-c0-0-2
Degree $2$
Conductor $1664$
Sign $0.957 - 0.289i$
Analytic cond. $0.830444$
Root an. cond. $0.911287$
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  + (1 + i)5-s + 9-s − i·13-s − 2i·17-s + i·25-s + 2i·29-s + (−1 + i)37-s + (−1 + i)41-s + (1 + i)45-s − i·49-s + (1 − i)65-s + (−1 − i)73-s + 81-s + (2 − 2i)85-s + (1 + i)89-s + ⋯
L(s)  = 1  + (1 + i)5-s + 9-s − i·13-s − 2i·17-s + i·25-s + 2i·29-s + (−1 + i)37-s + (−1 + i)41-s + (1 + i)45-s − i·49-s + (1 − i)65-s + (−1 − i)73-s + 81-s + (2 − 2i)85-s + (1 + i)89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1664\)    =    \(2^{7} \cdot 13\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1664} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1664,\ (\ :0),\ 0.957 - 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.420023503\)
\(L(\frac12)\) \(\approx\) \(1.420023503\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + iT \)
good3 \( 1 - T^{2} \)
5 \( 1 + (-1 - i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 + 2iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + (1 - i)T - iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-1 - i)T + iT^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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

−9.772287455075128243736255350548, −8.992016577640388159166194056060, −7.86353029367039957700413413587, −6.91842670664894703217684565221, −6.70538868122008566717993514199, −5.41232773629793173819227472901, −4.88478663307230829711424231106, −3.37277236302872262156437519393, −2.72456432773617871993018643651, −1.47650125863509594565976935798, 1.49032892334044718474348687978, 2.08019541926203373167180916363, 3.90535762723991259579051981908, 4.43621754241147280583428735928, 5.53481519413186673791138207472, 6.18307805750929971068001391010, 7.04317955137419410965198368206, 8.100656429660819117479706660719, 8.817774709759605093641099423991, 9.522468407946930247295630601618

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