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Properties

Label 2-891-99.95-c1-0-35
Degree $2$
Conductor $891$
Sign $0.921 + 0.388i$
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  + (−0.0265 + 0.252i)2-s + (1.89 + 0.402i)4-s + (2.90 − 0.305i)5-s + (−2.02 − 1.81i)7-s + (−0.308 + 0.949i)8-s + 0.741i·10-s + (0.283 − 3.30i)11-s + (−1.16 − 2.60i)13-s + (0.512 − 0.461i)14-s + (3.30 + 1.47i)16-s + (3.60 − 2.62i)17-s + (−1.81 − 0.590i)19-s + (5.62 + 0.591i)20-s + (0.826 + 0.159i)22-s + (0.706 + 0.408i)23-s + ⋯
L(s)  = 1  + (−0.0187 + 0.178i)2-s + (0.946 + 0.201i)4-s + (1.29 − 0.136i)5-s + (−0.763 − 0.687i)7-s + (−0.109 + 0.335i)8-s + 0.234i·10-s + (0.0855 − 0.996i)11-s + (−0.322 − 0.723i)13-s + (0.136 − 0.123i)14-s + (0.826 + 0.367i)16-s + (0.875 − 0.636i)17-s + (−0.416 − 0.135i)19-s + (1.25 + 0.132i)20-s + (0.176 + 0.0339i)22-s + (0.147 + 0.0851i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.921 + 0.388i$
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: \((2,\ 891,\ (\ :1/2),\ 0.921 + 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19846 - 0.445086i\)
\(L(\frac12)\) \(\approx\) \(2.19846 - 0.445086i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.283 + 3.30i)T \)
good2 \( 1 + (0.0265 - 0.252i)T + (-1.95 - 0.415i)T^{2} \)
5 \( 1 + (-2.90 + 0.305i)T + (4.89 - 1.03i)T^{2} \)
7 \( 1 + (2.02 + 1.81i)T + (0.731 + 6.96i)T^{2} \)
13 \( 1 + (1.16 + 2.60i)T + (-8.69 + 9.66i)T^{2} \)
17 \( 1 + (-3.60 + 2.62i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.81 + 0.590i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.706 - 0.408i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.38 + 7.09i)T + (-3.03 - 28.8i)T^{2} \)
31 \( 1 + (5.46 - 2.43i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (-1.83 - 5.65i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-5.64 - 6.26i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (10.2 - 5.93i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.60 - 7.54i)T + (-42.9 + 19.1i)T^{2} \)
53 \( 1 + (6.14 - 8.45i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.0193 - 0.0912i)T + (-53.8 - 23.9i)T^{2} \)
61 \( 1 + (-0.872 + 1.95i)T + (-40.8 - 45.3i)T^{2} \)
67 \( 1 + (-0.703 + 1.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.12 + 2.91i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.67 - 0.543i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.09 - 0.745i)T + (77.2 + 16.4i)T^{2} \)
83 \( 1 + (-7.15 - 3.18i)T + (55.5 + 61.6i)T^{2} \)
89 \( 1 - 2.06iT - 89T^{2} \)
97 \( 1 + (0.174 - 1.66i)T + (-94.8 - 20.1i)T^{2} \)
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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

−10.04086711012153554024659258971, −9.476064328773507754514132809389, −8.244350393183511025497604440084, −7.48539663068897831208794979493, −6.31036283082190460645697620039, −6.12515253002331120625865323102, −4.97995967933132724604390063909, −3.32490517924418589122364140512, −2.64963303861764265971370716751, −1.13795311269743161455744005816, 1.76707070188278327710075927621, 2.34097197145713863064769674643, 3.54607284213184899784770596557, 5.16414016419734175540578458265, 5.96018798462193911107269170271, 6.63981108116167214247613247864, 7.32973526441419351382465749485, 8.764470234353723556312577168694, 9.590703580329858084023090488140, 10.10765931600591479399825397826

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