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Properties

Label 2-891-99.31-c1-0-34
Degree $2$
Conductor $891$
Sign $0.990 + 0.140i$
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  + (2.40 − 0.511i)2-s + (3.70 − 1.64i)4-s + (0.968 + 0.205i)5-s + (−0.451 + 4.29i)7-s + (4.08 − 2.96i)8-s + 2.43·10-s + (0.155 − 3.31i)11-s + (2.42 + 2.68i)13-s + (1.11 + 10.5i)14-s + (2.89 − 3.21i)16-s + (−0.816 − 2.51i)17-s + (3.09 − 2.24i)19-s + (3.92 − 0.834i)20-s + (−1.31 − 8.05i)22-s + (1.72 − 2.98i)23-s + ⋯
L(s)  = 1  + (1.70 − 0.361i)2-s + (1.85 − 0.824i)4-s + (0.432 + 0.0920i)5-s + (−0.170 + 1.62i)7-s + (1.44 − 1.04i)8-s + 0.770·10-s + (0.0469 − 0.998i)11-s + (0.671 + 0.745i)13-s + (0.297 + 2.82i)14-s + (0.723 − 0.803i)16-s + (−0.197 − 0.609i)17-s + (0.710 − 0.516i)19-s + (0.877 − 0.186i)20-s + (−0.281 − 1.71i)22-s + (0.359 − 0.623i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.990 + 0.140i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.990 + 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.42418 - 0.312123i\)
\(L(\frac12)\) \(\approx\) \(4.42418 - 0.312123i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.155 + 3.31i)T \)
good2 \( 1 + (-2.40 + 0.511i)T + (1.82 - 0.813i)T^{2} \)
5 \( 1 + (-0.968 - 0.205i)T + (4.56 + 2.03i)T^{2} \)
7 \( 1 + (0.451 - 4.29i)T + (-6.84 - 1.45i)T^{2} \)
13 \( 1 + (-2.42 - 2.68i)T + (-1.35 + 12.9i)T^{2} \)
17 \( 1 + (0.816 + 2.51i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.09 + 2.24i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.72 + 2.98i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.08 - 10.3i)T + (-28.3 - 6.02i)T^{2} \)
31 \( 1 + (1.12 + 1.25i)T + (-3.24 + 30.8i)T^{2} \)
37 \( 1 + (2.61 + 1.90i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.24 + 11.8i)T + (-40.1 + 8.52i)T^{2} \)
43 \( 1 + (-1.06 - 1.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.92 + 1.74i)T + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (-2.93 + 9.01i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.91 - 2.63i)T + (39.4 - 43.8i)T^{2} \)
61 \( 1 + (3.21 - 3.57i)T + (-6.37 - 60.6i)T^{2} \)
67 \( 1 + (0.427 - 0.739i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.16 - 3.59i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (9.22 + 6.70i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (6.00 - 1.27i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (2.24 - 2.49i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + (-0.363 + 0.0772i)T + (88.6 - 39.4i)T^{2} \)
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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

−10.47816670841153272496103047246, −9.098389659957915538880545821012, −8.751535908361366577935742036509, −7.05203255987818527624130482980, −6.21857358287926170952894117388, −5.58364717309963409274586113271, −4.95691965399885821795442019260, −3.61169253743977768494079199209, −2.82506193142502707552818805080, −1.88798523699328536316986320375, 1.58212490171419412707799817306, 3.17082825136683945634482695062, 3.97955608428232424236535543204, 4.68056085876662608124649361239, 5.76927335604584504447847276979, 6.42643179221788652259541841331, 7.42187035951023263550635236665, 7.87135028197470768330131227626, 9.601061112522684990626619581954, 10.26128479348517042014509573644

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