Properties

Label 2-12e3-36.23-c1-0-3
Degree $2$
Conductor $1728$
Sign $-0.642 - 0.766i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 0.866i)5-s + (−1.5 + 0.866i)7-s + (1.5 + 2.59i)11-s + (−2.5 + 4.33i)13-s − 6.92i·17-s + 3.46i·19-s + (−4.5 + 7.79i)23-s + (−1 − 1.73i)25-s + (1.5 − 0.866i)29-s + (−4.5 − 2.59i)31-s − 3·35-s − 2·37-s + (4.5 + 2.59i)41-s + (−4.5 + 2.59i)43-s + (−1.5 − 2.59i)47-s + ⋯
L(s)  = 1  + (0.670 + 0.387i)5-s + (−0.566 + 0.327i)7-s + (0.452 + 0.783i)11-s + (−0.693 + 1.20i)13-s − 1.68i·17-s + 0.794i·19-s + (−0.938 + 1.62i)23-s + (−0.200 − 0.346i)25-s + (0.278 − 0.160i)29-s + (−0.808 − 0.466i)31-s − 0.507·35-s − 0.328·37-s + (0.702 + 0.405i)41-s + (−0.686 + 0.396i)43-s + (−0.218 − 0.378i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.142691925\)
\(L(\frac12)\) \(\approx\) \(1.142691925\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.5 - 0.866i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.92iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 0.866i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.5 + 2.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.5 - 2.59i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.5 + 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (7.5 - 4.33i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.5 - 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.592472324899106747389898794433, −9.248718834402307276561432462140, −7.86759293095456980613272584971, −7.13208262648442434224892494159, −6.45872946180404446083530780377, −5.64682209067740815230690387265, −4.69797866011886189751019091636, −3.70139740425310828969702479608, −2.53473145856749289354138095208, −1.73917767817959871792773412354, 0.40191272026119507821178445512, 1.81786518851975759053170609093, 3.02695033543572044774826826771, 3.92740572685284767664605056745, 5.01798266307769108217538464752, 5.91177371719511057388272379425, 6.43153746549980052045646729457, 7.46634096739167109058106626830, 8.440258879515500557185043099971, 8.914850832908453786964187646124

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