Properties

Label 2-891-99.95-c1-0-30
Degree $2$
Conductor $891$
Sign $-0.786 + 0.617i$
Analytic cond. $7.11467$
Root an. cond. $2.66733$
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  + (0.254 − 2.42i)2-s + (−3.84 − 0.818i)4-s + (3.77 − 0.396i)5-s + (0.273 + 0.246i)7-s + (−1.45 + 4.48i)8-s − 9.24i·10-s + (2.75 + 1.84i)11-s + (−0.385 − 0.866i)13-s + (0.665 − 0.599i)14-s + (3.30 + 1.47i)16-s + (2.77 − 2.01i)17-s + (4.05 + 1.31i)19-s + (−14.8 − 1.56i)20-s + (5.17 − 6.20i)22-s + (−4.30 − 2.48i)23-s + ⋯
L(s)  = 1  + (0.180 − 1.71i)2-s + (−1.92 − 0.409i)4-s + (1.68 − 0.177i)5-s + (0.103 + 0.0930i)7-s + (−0.515 + 1.58i)8-s − 2.92i·10-s + (0.830 + 0.557i)11-s + (−0.106 − 0.240i)13-s + (0.177 − 0.160i)14-s + (0.826 + 0.367i)16-s + (0.673 − 0.489i)17-s + (0.929 + 0.302i)19-s + (−3.32 − 0.349i)20-s + (1.10 − 1.32i)22-s + (−0.897 − 0.517i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.786 + 0.617i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (458, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ -0.786 + 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.719673 - 2.08124i\)
\(L(\frac12)\) \(\approx\) \(0.719673 - 2.08124i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (-2.75 - 1.84i)T \)
good2 \( 1 + (-0.254 + 2.42i)T + (-1.95 - 0.415i)T^{2} \)
5 \( 1 + (-3.77 + 0.396i)T + (4.89 - 1.03i)T^{2} \)
7 \( 1 + (-0.273 - 0.246i)T + (0.731 + 6.96i)T^{2} \)
13 \( 1 + (0.385 + 0.866i)T + (-8.69 + 9.66i)T^{2} \)
17 \( 1 + (-2.77 + 2.01i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.05 - 1.31i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.30 + 2.48i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.66 - 1.84i)T + (-3.03 - 28.8i)T^{2} \)
31 \( 1 + (-3.21 + 1.42i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (2.21 + 6.83i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.81 + 2.02i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (1.63 - 0.943i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.00428 + 0.0201i)T + (-42.9 + 19.1i)T^{2} \)
53 \( 1 + (3.25 - 4.48i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.37 - 6.47i)T + (-53.8 - 23.9i)T^{2} \)
61 \( 1 + (3.96 - 8.90i)T + (-40.8 - 45.3i)T^{2} \)
67 \( 1 + (2.23 - 3.86i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.06 - 8.35i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.18 - 1.35i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.8 + 1.14i)T + (77.2 + 16.4i)T^{2} \)
83 \( 1 + (8.38 + 3.73i)T + (55.5 + 61.6i)T^{2} \)
89 \( 1 + 3.04iT - 89T^{2} \)
97 \( 1 + (-1.57 + 15.0i)T + (-94.8 - 20.1i)T^{2} \)
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

−10.01227953598626329566964725538, −9.386361644315387579441449147091, −8.701647994120389880788010138332, −7.20416524407369550738619553479, −5.93093552702309900529861285235, −5.22136233146373007737201547022, −4.18865226617555717026263430161, −2.98398572652778916230384923831, −2.03262503735435045687481056834, −1.20361100840009606270786374729, 1.59013871781476403636401569323, 3.33260137649721837462129451024, 4.70609346881987407918893332590, 5.57220592765045289023375848970, 6.18082564096992469711714671188, 6.73951642316776494920351282455, 7.79108194836004197090993873082, 8.628987512002492127030947261497, 9.547235623865241200876971847166, 9.893497928688817843288466816771

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