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Properties

Label 2-891-33.32-c1-0-16
Degree $2$
Conductor $891$
Sign $1$
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  − 2·4-s − 0.939i·5-s + 3.31i·11-s + 4·16-s + 1.87i·20-s − 3.31i·23-s + 4.11·25-s + 6.11·31-s + 12.1·37-s − 6.63i·44-s − 13.7i·47-s + 7·49-s + 11.8i·53-s + 3.11·55-s + 14.6i·59-s + ⋯
L(s)  = 1  − 4-s − 0.420i·5-s + 1.00i·11-s + 16-s + 0.420i·20-s − 0.691i·23-s + 0.823·25-s + 1.09·31-s + 1.99·37-s − 1.00i·44-s − 1.99i·47-s + 49-s + 1.62i·53-s + 0.420·55-s + 1.90i·59-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $1$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (890, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20930\)
\(L(\frac12)\) \(\approx\) \(1.20930\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.31iT \)
good2 \( 1 + 2T^{2} \)
5 \( 1 + 0.939iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 3.31iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 6.11T + 31T^{2} \)
37 \( 1 - 12.1T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 13.7iT - 47T^{2} \)
53 \( 1 - 11.8iT - 53T^{2} \)
59 \( 1 - 14.6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 - 0.116T + 97T^{2} \)
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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

−9.988529922059907118071337913716, −9.273493641523635836020925291894, −8.544528680950892460326095058264, −7.75187537201464943266734075886, −6.71109643924283595376348953142, −5.57461224004070071035271830005, −4.66727546044483892277171332605, −4.08569270821155434084443683803, −2.59873329176481256292351133490, −0.939577552880955272023341599912, 0.888547202080832635019228880006, 2.80313156871787134858264730748, 3.76291130746826121004391761659, 4.77909097879838579292599750801, 5.73547929252477728782997258710, 6.58688959157858818618333595203, 7.82186833283337720954730381103, 8.414843186441910994549763637020, 9.344448731133134341151082844620, 9.975279304637787624650287887877

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