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Properties

Label 2-891-99.74-c1-0-18
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: $\chi_{891} (107, \cdot )$
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

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

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