↑

Properties

Label 2-855-95.4-c1-0-30
Degree $2$
Conductor $855$
Sign $0.545 + 0.837i$
Analytic cond. $6.82720$
Root an. cond. $2.61289$
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  + (−0.358 + 0.984i)2-s + (0.691 + 0.580i)4-s + (−2.23 − 0.0926i)5-s + (−2.37 − 1.37i)7-s + (−2.63 + 1.52i)8-s + (0.891 − 2.16i)10-s + (0.416 + 0.721i)11-s + (0.601 + 0.106i)13-s + (2.19 − 1.84i)14-s + (−0.239 − 1.35i)16-s + (1.65 − 4.54i)17-s + (−4.35 − 0.175i)19-s + (−1.49 − 1.36i)20-s + (−0.859 + 0.151i)22-s + (2.41 − 2.87i)23-s + ⋯
L(s)  = 1  + (−0.253 + 0.695i)2-s + (0.345 + 0.290i)4-s + (−0.999 − 0.0414i)5-s + (−0.896 − 0.517i)7-s + (−0.930 + 0.537i)8-s + (0.281 − 0.684i)10-s + (0.125 + 0.217i)11-s + (0.166 + 0.0294i)13-s + (0.587 − 0.493i)14-s + (−0.0598 − 0.339i)16-s + (0.401 − 1.10i)17-s + (−0.999 − 0.0402i)19-s + (−0.333 − 0.304i)20-s + (−0.183 + 0.0322i)22-s + (0.502 − 0.599i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.545 + 0.837i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.545 + 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.497059 - 0.269387i\)
\(L(\frac12)\) \(\approx\) \(0.497059 - 0.269387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (2.23 + 0.0926i)T \)
19 \( 1 + (4.35 + 0.175i)T \)
good2 \( 1 + (0.358 - 0.984i)T + (-1.53 - 1.28i)T^{2} \)
7 \( 1 + (2.37 + 1.37i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.416 - 0.721i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.601 - 0.106i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.65 + 4.54i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-2.41 + 2.87i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.73 + 1.35i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.46 + 5.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.33iT - 37T^{2} \)
41 \( 1 + (0.923 + 5.23i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (6.72 + 8.01i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (1.16 + 3.19i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (8.78 - 10.4i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (9.41 + 3.42i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-6.94 - 5.83i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-3.73 - 10.2i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-0.519 + 0.435i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (6.90 - 1.21i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.604 + 3.42i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (4.30 + 2.48i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.02 - 5.79i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-4.25 + 11.6i)T + (-74.3 - 62.3i)T^{2} \)
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

−9.975900395957940022664447705624, −8.926756433431531293501292220349, −8.268184131383736988487071684252, −7.28306084704202489395192236799, −6.90713389479550088205593719111, −5.98436866101337031147396775605, −4.61882238746935744590625398842, −3.60347918135262767581297045582, −2.65138816053727617043767656517, −0.30329357668735605240707793977, 1.39476071229371428877235783786, 2.96371289404995203977102631410, 3.50125149892036986512176984273, 4.87264156051344986871017747329, 6.33338924581321281098899165782, 6.55285793524627107005983842223, 7.994936216862734708858404951965, 8.695960768879525078589405330105, 9.642717263811369785083464579219, 10.39313732912790908644633821861

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