↑

Properties

Label 2-864-72.13-c1-0-6
Degree $2$
Conductor $864$
Sign $0.808 + 0.588i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
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  + (−0.602 − 0.348i)5-s + (−0.795 − 1.37i)7-s + (−2.37 + 1.36i)11-s + (4.76 + 2.75i)13-s + 5.65·17-s − 0.963i·19-s + (3.28 − 5.69i)23-s + (−2.25 − 3.91i)25-s + (2.85 − 1.64i)29-s + (3.69 − 6.40i)31-s + 1.10i·35-s − 6.25i·37-s + (0.931 − 1.61i)41-s + (−2.99 + 1.73i)43-s + (3.85 + 6.67i)47-s + ⋯
L(s)  = 1  + (−0.269 − 0.155i)5-s + (−0.300 − 0.520i)7-s + (−0.715 + 0.412i)11-s + (1.32 + 0.763i)13-s + 1.37·17-s − 0.221i·19-s + (0.685 − 1.18i)23-s + (−0.451 − 0.782i)25-s + (0.529 − 0.305i)29-s + (0.664 − 1.15i)31-s + 0.187i·35-s − 1.02i·37-s + (0.145 − 0.252i)41-s + (−0.457 + 0.263i)43-s + (0.562 + 0.974i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.808 + 0.588i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.808 + 0.588i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38865 - 0.452162i\)
\(L(\frac12)\) \(\approx\) \(1.38865 - 0.452162i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.602 + 0.348i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.795 + 1.37i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.37 - 1.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.76 - 2.75i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.65T + 17T^{2} \)
19 \( 1 + 0.963iT - 19T^{2} \)
23 \( 1 + (-3.28 + 5.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.85 + 1.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.69 + 6.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.25iT - 37T^{2} \)
41 \( 1 + (-0.931 + 1.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.99 - 1.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.85 - 6.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.54iT - 53T^{2} \)
59 \( 1 + (-4.62 - 2.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.93 - 4.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.95 - 3.43i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 - 2.83T + 73T^{2} \)
79 \( 1 + (2.87 + 4.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.74 + 3.31i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.98T + 89T^{2} \)
97 \( 1 + (1.24 + 2.16i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.18452489900543871454476710315, −9.253625201679488099430509669743, −8.313407645157345455166794780537, −7.61418243714486025619282854092, −6.62220982056957570291177763659, −5.79118749765542292290891059971, −4.55850711954804268614237631849, −3.79344647170478548936574426077, −2.52517323021885875399291012398, −0.868672839909202483037483279761, 1.23197819192160658991060545261, 3.05769611424175296020241607649, 3.52741440895975450875913807097, 5.21132996240929352068305047613, 5.73094047979845253678608477466, 6.79608624151350461507061058263, 7.909168782086009038581428952443, 8.394448902047421666936945628251, 9.436604905375354863406207152635, 10.31129019927271271083532672002

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