Properties

Label 2-891-11.9-c1-0-16
Degree $2$
Conductor $891$
Sign $-0.244 - 0.969i$
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  + (−0.143 + 0.441i)2-s + (1.44 + 1.04i)4-s + (0.808 + 2.48i)5-s + (1.43 + 1.04i)7-s + (−1.42 + 1.03i)8-s − 1.21·10-s + (1.63 + 2.88i)11-s + (1.67 − 5.14i)13-s + (−0.667 + 0.484i)14-s + (0.851 + 2.62i)16-s + (0.343 + 1.05i)17-s + (−2.64 + 1.92i)19-s + (−1.44 + 4.44i)20-s + (−1.50 + 0.308i)22-s + 4.23·23-s + ⋯
L(s)  = 1  + (−0.101 + 0.311i)2-s + (0.721 + 0.524i)4-s + (0.361 + 1.11i)5-s + (0.543 + 0.394i)7-s + (−0.502 + 0.364i)8-s − 0.383·10-s + (0.493 + 0.869i)11-s + (0.463 − 1.42i)13-s + (−0.178 + 0.129i)14-s + (0.212 + 0.655i)16-s + (0.0832 + 0.256i)17-s + (−0.606 + 0.440i)19-s + (−0.322 + 0.993i)20-s + (−0.321 + 0.0657i)22-s + 0.883·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.244 - 0.969i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ -0.244 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23967 + 1.59185i\)
\(L(\frac12)\) \(\approx\) \(1.23967 + 1.59185i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 - 2.88i)T \)
good2 \( 1 + (0.143 - 0.441i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.808 - 2.48i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-1.43 - 1.04i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.67 + 5.14i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.343 - 1.05i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.64 - 1.92i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + (6.27 + 4.55i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.95 + 9.08i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.95 - 2.87i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.12 - 2.99i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 9.53T + 43T^{2} \)
47 \( 1 + (3.69 - 2.68i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.06 + 9.44i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.48 - 1.80i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.65 - 5.08i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 9.10T + 67T^{2} \)
71 \( 1 + (0.243 + 0.749i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.16 - 0.847i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.23 + 6.87i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.53 - 10.8i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + (5.82 - 17.9i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.37602090054680318697125712347, −9.677921458355012285477257272370, −8.321373067514679307820147462441, −7.891962693587400443004402598249, −6.86555980152018812871204417612, −6.27290664675266629176023715654, −5.36725217036398294232968129192, −3.85753750373881949938458813259, −2.83068572261751592686844788987, −1.93293928978700925461473330276, 1.07934157410784003969666564466, 1.81465442679631645824060193106, 3.37851362434878257217291907289, 4.63289363826713687491372113844, 5.41918863249622652967252149006, 6.49243310637094988174378134334, 7.12376643669846377656952843108, 8.606467227534221220167814795694, 8.955597359132639211635887861808, 9.835803548928658004221780338899

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