Properties

Label 2-864-32.13-c1-0-9
Degree $2$
Conductor $864$
Sign $0.498 - 0.866i$
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  + (−1.01 − 0.986i)2-s + (0.0539 + 1.99i)4-s + (−2.55 + 1.05i)5-s + (1.06 − 1.06i)7-s + (1.91 − 2.07i)8-s + (3.62 + 1.44i)10-s + (−0.641 − 1.54i)11-s + (0.200 + 0.0830i)13-s + (−2.12 + 0.0287i)14-s + (−3.99 + 0.215i)16-s + 2.47i·17-s + (4.02 + 1.66i)19-s + (−2.25 − 5.04i)20-s + (−0.877 + 2.20i)22-s + (−2.90 − 2.90i)23-s + ⋯
L(s)  = 1  + (−0.716 − 0.697i)2-s + (0.0269 + 0.999i)4-s + (−1.14 + 0.472i)5-s + (0.402 − 0.402i)7-s + (0.677 − 0.735i)8-s + (1.14 + 0.457i)10-s + (−0.193 − 0.467i)11-s + (0.0556 + 0.0230i)13-s + (−0.569 + 0.00767i)14-s + (−0.998 + 0.0538i)16-s + 0.599i·17-s + (0.923 + 0.382i)19-s + (−0.503 − 1.12i)20-s + (−0.187 + 0.469i)22-s + (−0.606 − 0.606i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.498 - 0.866i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.498 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.516240 + 0.298719i\)
\(L(\frac12)\) \(\approx\) \(0.516240 + 0.298719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.01 + 0.986i)T \)
3 \( 1 \)
good5 \( 1 + (2.55 - 1.05i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.06 + 1.06i)T - 7iT^{2} \)
11 \( 1 + (0.641 + 1.54i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.200 - 0.0830i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 2.47iT - 17T^{2} \)
19 \( 1 + (-4.02 - 1.66i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (2.90 + 2.90i)T + 23iT^{2} \)
29 \( 1 + (1.64 - 3.97i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 0.400T + 31T^{2} \)
37 \( 1 + (4.73 - 1.96i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.07 - 2.07i)T + 41iT^{2} \)
43 \( 1 + (-3.45 - 8.34i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 6.94iT - 47T^{2} \)
53 \( 1 + (-4.57 - 11.0i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (5.70 - 2.36i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.321 - 0.775i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (5.91 - 14.2i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-7.28 + 7.28i)T - 71iT^{2} \)
73 \( 1 + (7.84 + 7.84i)T + 73iT^{2} \)
79 \( 1 + 1.11iT - 79T^{2} \)
83 \( 1 + (-12.4 - 5.13i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-7.58 + 7.58i)T - 89iT^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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

−10.68761721291825928765539854389, −9.507707817045373613826368904344, −8.570353537570771572868790487957, −7.75523613643828431634749924744, −7.40803429096949442695992221589, −6.14432386427174079369244320382, −4.56855034645556399550421348618, −3.72355642696927100917431083470, −2.87697533234024377568850163220, −1.25604294351581013422682964574, 0.41264854725097819796434489900, 2.05732212618451686444085731268, 3.79554790141831862762534468898, 4.93416718177181340631445153613, 5.55870442016054714574784105604, 6.92035584037562232539058634084, 7.60273217816857741040814895318, 8.221514762765701639373670056899, 9.039018101941982897003299232473, 9.781410456406087776684112475288

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