↑

Properties

Label 2-864-72.11-c1-0-2
Degree $2$
Conductor $864$
Sign $0.282 - 0.959i$
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  + (1.60 + 2.78i)5-s + (−1.82 − 1.05i)7-s + (3.47 + 2.00i)11-s + (−0.341 + 0.197i)13-s − 1.20i·17-s + 1.62·19-s + (2.74 + 4.75i)23-s + (−2.68 + 4.64i)25-s + (−2.95 + 5.12i)29-s + (−3.34 + 1.93i)31-s − 6.77i·35-s + 10.8i·37-s + (1.23 − 0.715i)41-s + (1.21 − 2.10i)43-s + (0.792 − 1.37i)47-s + ⋯
L(s)  = 1  + (0.719 + 1.24i)5-s + (−0.688 − 0.397i)7-s + (1.04 + 0.605i)11-s + (−0.0948 + 0.0547i)13-s − 0.292i·17-s + 0.372·19-s + (0.572 + 0.990i)23-s + (−0.536 + 0.928i)25-s + (−0.549 + 0.950i)29-s + (−0.601 + 0.347i)31-s − 1.14i·35-s + 1.77i·37-s + (0.193 − 0.111i)41-s + (0.185 − 0.321i)43-s + (0.115 − 0.200i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.282 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.282 - 0.959i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.282 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31079 + 0.980660i\)
\(L(\frac12)\) \(\approx\) \(1.31079 + 0.980660i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.60 - 2.78i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.82 + 1.05i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.47 - 2.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.341 - 0.197i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.20iT - 17T^{2} \)
19 \( 1 - 1.62T + 19T^{2} \)
23 \( 1 + (-2.74 - 4.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.95 - 5.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.34 - 1.93i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 + (-1.23 + 0.715i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.21 + 2.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.792 + 1.37i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.07T + 53T^{2} \)
59 \( 1 + (2.29 - 1.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.18 + 4.72i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.60 - 4.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.69T + 71T^{2} \)
73 \( 1 - 9.49T + 73T^{2} \)
79 \( 1 + (1.53 + 0.886i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.30 + 0.755i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.2iT - 89T^{2} \)
97 \( 1 + (-5.84 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.18997143790244738597693607400, −9.661745796265934295943377529059, −8.927722479197955398921499487204, −7.38878244314042772235533449556, −6.91970754415291024557330335506, −6.23234824469354855944128123322, −5.12941401708829275155140129711, −3.73215886418353653026859785988, −2.96944929878961008578418829752, −1.61926358680699830471069551602, 0.842319227740148753866339046423, 2.21146336761158933947120826378, 3.62627906253282834743468816000, 4.67464530952224101029733196232, 5.78034803059317844874569096197, 6.20692544891447888724073445347, 7.46503178389385434938586711079, 8.658918734186798949646811483708, 9.146588785672335919397361505666, 9.673968070236356378612713372871

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