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Properties

Label 2-891-99.31-c1-0-22
Degree $2$
Conductor $891$
Sign $-0.0823 + 0.996i$
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  + (−0.604 + 0.128i)2-s + (−1.47 + 0.658i)4-s + (−2.56 − 0.544i)5-s + (−0.313 + 2.98i)7-s + (1.80 − 1.31i)8-s + 1.61·10-s + (−1.62 + 2.89i)11-s + (1.18 + 1.31i)13-s + (−0.193 − 1.84i)14-s + (1.24 − 1.37i)16-s + (−0.5 − 1.53i)17-s + (−4.73 + 3.44i)19-s + (4.14 − 0.880i)20-s + (0.611 − 1.95i)22-s + (1.73 − 3.00i)23-s + ⋯
L(s)  = 1  + (−0.427 + 0.0908i)2-s + (−0.739 + 0.329i)4-s + (−1.14 − 0.243i)5-s + (−0.118 + 1.12i)7-s + (0.639 − 0.464i)8-s + 0.511·10-s + (−0.490 + 0.871i)11-s + (0.327 + 0.363i)13-s + (−0.0517 − 0.492i)14-s + (0.310 − 0.344i)16-s + (−0.121 − 0.373i)17-s + (−1.08 + 0.789i)19-s + (0.926 − 0.196i)20-s + (0.130 − 0.417i)22-s + (0.361 − 0.626i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0823 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0823 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.0823 + 0.996i$
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: \((2,\ 891,\ (\ :1/2),\ -0.0823 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.129158 - 0.140272i\)
\(L(\frac12)\) \(\approx\) \(0.129158 - 0.140272i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 2.89i)T \)
good2 \( 1 + (0.604 - 0.128i)T + (1.82 - 0.813i)T^{2} \)
5 \( 1 + (2.56 + 0.544i)T + (4.56 + 2.03i)T^{2} \)
7 \( 1 + (0.313 - 2.98i)T + (-6.84 - 1.45i)T^{2} \)
13 \( 1 + (-1.18 - 1.31i)T + (-1.35 + 12.9i)T^{2} \)
17 \( 1 + (0.5 + 1.53i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.73 + 3.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.467 + 4.44i)T + (-28.3 - 6.02i)T^{2} \)
31 \( 1 + (-1.90 - 2.12i)T + (-3.24 + 30.8i)T^{2} \)
37 \( 1 + (0.190 + 0.138i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.24 + 11.8i)T + (-40.1 + 8.52i)T^{2} \)
43 \( 1 + (3.11 + 5.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.47 + 0.658i)T + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (-2.97 + 9.14i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (9.43 - 4.20i)T + (39.4 - 43.8i)T^{2} \)
61 \( 1 + (-5.25 + 5.83i)T + (-6.37 - 60.6i)T^{2} \)
67 \( 1 + (-4.78 + 8.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.71 - 5.29i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.61 - 1.90i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (9.26 - 1.96i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (0.473 - 0.526i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + 0.527T + 89T^{2} \)
97 \( 1 + (-13.7 + 2.91i)T + (88.6 - 39.4i)T^{2} \)
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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

−9.764005540373157070210786937710, −8.778303470826647471146292704273, −8.424192542421328370643366328519, −7.61445074740669815319221568937, −6.66347382022731745795514615633, −5.33522187504087730866465798206, −4.45556347238062537471262722396, −3.67386433726863690860800751752, −2.19995030019269545701395086962, −0.13348266805844364110315596623, 1.05609681434717769547372638030, 3.14185771986428678549231269892, 4.06268107698335352186042070140, 4.81205185625514524131697707585, 6.08371863571627781837163342189, 7.17306666987309896381071507577, 8.021310208111482729569895734785, 8.486174636722306820470730288460, 9.518872189098835856761758935152, 10.56398814095118796645117841630

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