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Properties

Label 2-891-99.95-c1-0-23
Degree $2$
Conductor $891$
Sign $-0.396 - 0.918i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.189 + 1.80i)2-s + (−1.25 − 0.265i)4-s + (4.27 − 0.448i)5-s + (2.14 + 1.92i)7-s + (−0.403 + 1.24i)8-s + 7.77i·10-s + (−0.650 − 3.25i)11-s + (0.272 + 0.612i)13-s + (−3.87 + 3.49i)14-s + (−4.49 − 2.00i)16-s + (−4.83 + 3.51i)17-s + (2.08 + 0.677i)19-s + (−5.45 − 0.573i)20-s + (5.97 − 0.556i)22-s + (−0.381 − 0.220i)23-s + ⋯
L(s)  = 1  + (−0.133 + 1.27i)2-s + (−0.625 − 0.132i)4-s + (1.91 − 0.200i)5-s + (0.809 + 0.728i)7-s + (−0.142 + 0.439i)8-s + 2.45i·10-s + (−0.196 − 0.980i)11-s + (0.0756 + 0.169i)13-s + (−1.03 + 0.933i)14-s + (−1.12 − 0.500i)16-s + (−1.17 + 0.852i)17-s + (0.478 + 0.155i)19-s + (−1.22 − 0.128i)20-s + (1.27 − 0.118i)22-s + (−0.0795 − 0.0459i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.396 - 0.918i$
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: \((2,\ 891,\ (\ :1/2),\ -0.396 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21433 + 1.84619i\)
\(L(\frac12)\) \(\approx\) \(1.21433 + 1.84619i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.650 + 3.25i)T \)
good2 \( 1 + (0.189 - 1.80i)T + (-1.95 - 0.415i)T^{2} \)
5 \( 1 + (-4.27 + 0.448i)T + (4.89 - 1.03i)T^{2} \)
7 \( 1 + (-2.14 - 1.92i)T + (0.731 + 6.96i)T^{2} \)
13 \( 1 + (-0.272 - 0.612i)T + (-8.69 + 9.66i)T^{2} \)
17 \( 1 + (4.83 - 3.51i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.08 - 0.677i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.381 + 0.220i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.110 - 0.122i)T + (-3.03 - 28.8i)T^{2} \)
31 \( 1 + (-6.42 + 2.86i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (-0.630 - 1.94i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (6.17 + 6.85i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (1.76 - 1.01i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.848 - 3.99i)T + (-42.9 + 19.1i)T^{2} \)
53 \( 1 + (2.37 - 3.26i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.101 - 0.479i)T + (-53.8 - 23.9i)T^{2} \)
61 \( 1 + (3.01 - 6.76i)T + (-40.8 - 45.3i)T^{2} \)
67 \( 1 + (-5.41 + 9.37i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.31 + 5.93i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.36 - 1.41i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.81 - 0.610i)T + (77.2 + 16.4i)T^{2} \)
83 \( 1 + (7.90 + 3.51i)T + (55.5 + 61.6i)T^{2} \)
89 \( 1 + 13.4iT - 89T^{2} \)
97 \( 1 + (0.881 - 8.38i)T + (-94.8 - 20.1i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.22369241574128990856233958970, −9.110818714672985678090386746755, −8.718407017310038769135482693681, −7.976353993492305544353745462282, −6.63761574124863622888606348804, −6.07983191216588383936424820894, −5.50183782096769065093818261065, −4.72558653910837220300452245009, −2.66219073713171785080510272874, −1.73220577181944428739687986431, 1.25797345287971501591823043909, 2.09782583736017322036796286836, 2.89905537857537280669596153927, 4.46342535955398063768717052793, 5.19283495367059778249685073843, 6.52490109193567148330450321542, 7.09829487223820981911878984332, 8.538856365121041146700819221796, 9.594188186041549967369878309505, 9.872934187144306268451770987197

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