↑

Properties

Label 2-12e3-24.5-c0-0-1
Degree $2$
Conductor $1728$
Sign $0.965 - 0.258i$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
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  + 1.73·7-s + 1.73i·13-s − i·19-s − 25-s + 1.73i·37-s − 2i·43-s + 1.99·49-s − 1.73i·61-s + i·67-s − 73-s + 1.73·79-s + 2.99i·91-s + 97-s − 1.73·103-s + ⋯
L(s)  = 1  + 1.73·7-s + 1.73i·13-s − i·19-s − 25-s + 1.73i·37-s − 2i·43-s + 1.99·49-s − 1.73i·61-s + i·67-s − 73-s + 1.73·79-s + 2.99i·91-s + 97-s − 1.73·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.965 - 0.258i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ 0.965 - 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.346821377\)
\(L(\frac12)\) \(\approx\) \(1.346821377\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 + T^{2} \)
7 \( 1 - 1.73T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 1.73iT - T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - 1.73T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T + T^{2} \)
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

−9.392192049554450542858585355197, −8.726559366612716346686431248580, −8.032262626150231921068616874432, −7.19853684935381494017709774935, −6.45943846897056164066551919052, −5.26162929618539286342752890034, −4.65168210636875656123279438289, −3.87590463956139070015727376229, −2.31466449259314218333480453223, −1.53209581930402943391581363298, 1.27374928404865229311054650625, 2.38737495484767966059098893467, 3.63427805708805487319670145154, 4.59546122557385217629789820771, 5.45725681129531527842857408575, 6.00036035500522940460046212826, 7.49991922128041286452351276793, 7.86786542220392293581463568578, 8.450131488255101074234254138421, 9.485588673504898886092293388365

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