↑

Properties

Label 2-855-95.83-c0-0-1
Degree $2$
Conductor $855$
Sign $0.875 + 0.483i$
Analytic cond. $0.426700$
Root an. cond. $0.653223$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (0.258 − 0.965i)5-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)10-s + 1.41·11-s + (−1.36 + 0.366i)13-s + (−0.5 − 0.866i)16-s + (0.965 + 0.258i)17-s + (0.866 − 0.5i)19-s + (1.36 + 0.366i)22-s + (−0.866 − 0.499i)25-s − 1.41·26-s − 31-s + (0.866 + 0.499i)34-s + (−1 + i)37-s + (0.965 − 0.258i)38-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.258 − 0.965i)5-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)10-s + 1.41·11-s + (−1.36 + 0.366i)13-s + (−0.5 − 0.866i)16-s + (0.965 + 0.258i)17-s + (0.866 − 0.5i)19-s + (1.36 + 0.366i)22-s + (−0.866 − 0.499i)25-s − 1.41·26-s − 31-s + (0.866 + 0.499i)34-s + (−1 + i)37-s + (0.965 − 0.258i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.875 + 0.483i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ 0.875 + 0.483i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.498532812\)
\(L(\frac12)\) \(\approx\) \(1.498532812\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.258 + 0.965i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.866 + 0.5i)T^{2} \)
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

−9.971330925782709655176186921153, −9.488352472572681854448272115064, −8.841058995898937510807575423672, −7.59140959953080191282424901665, −6.64452454918356941030622416202, −5.73084196722795640988328968352, −4.93075355145186209315449992929, −4.27310005272983387642983088924, −3.16060761744082939513041540492, −1.37216897434396377208484966373, 2.11410802975065682369895983258, 3.28935508522606319732484280058, 3.88012258770612556872780556326, 5.21382242880974775139431325829, 5.81322557571937828897831313455, 6.98826800408513590588320603964, 7.62564686005516533763596787240, 8.981613618776287363108585414717, 9.661719704399778121914722898836, 10.52784971346913530876232183972

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