Properties

Label 2-864-32.13-c1-0-54
Degree $2$
Conductor $864$
Sign $0.993 - 0.109i$
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  + (1.33 + 0.460i)2-s + (1.57 + 1.23i)4-s + (2.37 − 0.982i)5-s + (2.29 − 2.29i)7-s + (1.54 + 2.37i)8-s + (3.62 − 0.222i)10-s + (−0.00833 − 0.0201i)11-s + (−4.05 − 1.67i)13-s + (4.12 − 2.01i)14-s + (0.969 + 3.88i)16-s − 1.80i·17-s + (0.650 + 0.269i)19-s + (4.94 + 1.37i)20-s + (−0.00188 − 0.0307i)22-s + (−1.25 − 1.25i)23-s + ⋯
L(s)  = 1  + (0.945 + 0.325i)2-s + (0.788 + 0.615i)4-s + (1.06 − 0.439i)5-s + (0.867 − 0.867i)7-s + (0.544 + 0.838i)8-s + (1.14 − 0.0702i)10-s + (−0.00251 − 0.00606i)11-s + (−1.12 − 0.465i)13-s + (1.10 − 0.537i)14-s + (0.242 + 0.970i)16-s − 0.437i·17-s + (0.149 + 0.0617i)19-s + (1.10 + 0.306i)20-s + (−0.000401 − 0.00655i)22-s + (−0.262 − 0.262i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.993 - 0.109i$
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: \((2,\ 864,\ (\ :1/2),\ 0.993 - 0.109i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.46131 + 0.190134i\)
\(L(\frac12)\) \(\approx\) \(3.46131 + 0.190134i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 - 0.460i)T \)
3 \( 1 \)
good5 \( 1 + (-2.37 + 0.982i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.29 + 2.29i)T - 7iT^{2} \)
11 \( 1 + (0.00833 + 0.0201i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (4.05 + 1.67i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 1.80iT - 17T^{2} \)
19 \( 1 + (-0.650 - 0.269i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.25 + 1.25i)T + 23iT^{2} \)
29 \( 1 + (0.655 - 1.58i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 1.65T + 31T^{2} \)
37 \( 1 + (11.0 - 4.57i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.27 - 3.27i)T + 41iT^{2} \)
43 \( 1 + (2.93 + 7.08i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 9.31iT - 47T^{2} \)
53 \( 1 + (-5.35 - 12.9i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-8.36 + 3.46i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.0511 - 0.123i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (2.29 - 5.54i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-10.9 + 10.9i)T - 71iT^{2} \)
73 \( 1 + (-5.54 - 5.54i)T + 73iT^{2} \)
79 \( 1 + 1.71iT - 79T^{2} \)
83 \( 1 + (-8.31 - 3.44i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (8.03 - 8.03i)T - 89iT^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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

−10.32965791489597647867247413368, −9.398084456724612599526986727877, −8.247848053030737368911108731744, −7.45978873792059424111981466010, −6.69919270533552988906752880470, −5.46741722855349982596415192386, −5.05234479053000866736459557985, −4.06565237793292283456143498959, −2.68625008745360994141709404609, −1.54228567679525379145183917471, 1.92514066753935643171027116615, 2.33246033682868463852213623049, 3.75550636352838985575722214416, 5.08109760083879219144315141529, 5.48167496221023044136295816187, 6.48871884178715466395776343849, 7.30785990216230681061328878712, 8.531147404377845373628156654574, 9.623795090752524758054699845509, 10.18558012966831770130640764054

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