Properties

Label 2-891-11.9-c1-0-21
Degree $2$
Conductor $891$
Sign $0.989 - 0.143i$
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  + (0.330 − 1.01i)2-s + (0.691 + 0.502i)4-s + (1.08 + 3.32i)5-s + (−0.632 − 0.459i)7-s + (2.47 − 1.79i)8-s + 3.74·10-s + (3.03 − 1.32i)11-s + (−0.274 + 0.846i)13-s + (−0.677 + 0.492i)14-s + (−0.482 − 1.48i)16-s + (−0.720 − 2.21i)17-s + (−2.36 + 1.71i)19-s + (−0.924 + 2.84i)20-s + (−0.347 − 3.53i)22-s + 1.15·23-s + ⋯
L(s)  = 1  + (0.233 − 0.719i)2-s + (0.345 + 0.251i)4-s + (0.483 + 1.48i)5-s + (−0.239 − 0.173i)7-s + (0.873 − 0.634i)8-s + 1.18·10-s + (0.916 − 0.400i)11-s + (−0.0762 + 0.234i)13-s + (−0.181 + 0.131i)14-s + (−0.120 − 0.371i)16-s + (−0.174 − 0.537i)17-s + (−0.541 + 0.393i)19-s + (−0.206 + 0.635i)20-s + (−0.0741 − 0.753i)22-s + 0.241·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.989 - 0.143i$
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: \((2,\ 891,\ (\ :1/2),\ 0.989 - 0.143i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30936 + 0.166407i\)
\(L(\frac12)\) \(\approx\) \(2.30936 + 0.166407i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.03 + 1.32i)T \)
good2 \( 1 + (-0.330 + 1.01i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-1.08 - 3.32i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.632 + 0.459i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.274 - 0.846i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.720 + 2.21i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.36 - 1.71i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.15T + 23T^{2} \)
29 \( 1 + (-4.28 - 3.10i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.13 - 6.58i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-8.94 - 6.49i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-8.99 + 6.53i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 + (0.680 - 0.494i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.962 - 2.96i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.40 + 5.37i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.642 - 1.97i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 + (-3.29 - 10.1i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.17 + 4.48i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.559 + 1.72i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.97 + 9.15i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (-4.56 + 14.0i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.32025967399264216440571275237, −9.682820546693386470934585775253, −8.532095523445198065012962500865, −7.26389995541429485851198786275, −6.75997253023959259123630924016, −6.08099497956832991206765294249, −4.48653765514668197860241243432, −3.38242920177286807860244350619, −2.81622779722355070052479539238, −1.62106557014202649165266542131, 1.17137551235927916920436610627, 2.30777667988814810810582313125, 4.24506104653989197788478591151, 4.85063454948907530878998297628, 5.95599700869470616909495892520, 6.33452201951300611073263958184, 7.56068652281795499487014147465, 8.360964403184203957498501604207, 9.256702238271868305656399703001, 9.822892489933305261333092275518

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