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Properties

Label 2-891-11.3-c1-0-20
Degree $2$
Conductor $891$
Sign $0.730 - 0.682i$
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  + (−2.12 + 1.54i)2-s + (1.51 − 4.67i)4-s + (0.193 + 0.140i)5-s + (−0.366 + 1.12i)7-s + (2.36 + 7.28i)8-s − 0.629·10-s + (2.93 + 1.53i)11-s + (4.88 − 3.54i)13-s + (−0.963 − 2.96i)14-s + (−8.34 − 6.06i)16-s + (1.48 + 1.07i)17-s + (−1.00 − 3.10i)19-s + (0.952 − 0.691i)20-s + (−8.62 + 1.27i)22-s − 5.55·23-s + ⋯
L(s)  = 1  + (−1.50 + 1.09i)2-s + (0.759 − 2.33i)4-s + (0.0866 + 0.0629i)5-s + (−0.138 + 0.426i)7-s + (0.836 + 2.57i)8-s − 0.199·10-s + (0.886 + 0.463i)11-s + (1.35 − 0.984i)13-s + (−0.257 − 0.792i)14-s + (−2.08 − 1.51i)16-s + (0.359 + 0.261i)17-s + (−0.231 − 0.711i)19-s + (0.212 − 0.154i)20-s + (−1.83 + 0.271i)22-s − 1.15·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.730 - 0.682i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.730 - 0.682i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.736352 + 0.290425i\)
\(L(\frac12)\) \(\approx\) \(0.736352 + 0.290425i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.93 - 1.53i)T \)
good2 \( 1 + (2.12 - 1.54i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-0.193 - 0.140i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.366 - 1.12i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.88 + 3.54i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.48 - 1.07i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.00 + 3.10i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.55T + 23T^{2} \)
29 \( 1 + (-1.92 + 5.92i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.20 - 2.32i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.67 - 5.16i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.29 + 10.1i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 + (-0.511 - 1.57i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-6.99 + 5.07i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.21 - 3.73i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-8.67 - 6.30i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 9.22T + 67T^{2} \)
71 \( 1 + (5.77 + 4.19i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.172 - 0.531i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.57 - 2.59i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-12.4 - 9.06i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 0.0625T + 89T^{2} \)
97 \( 1 + (4.92 - 3.58i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.08659881970470604925212454361, −9.123207169677317284311363313882, −8.555031889801975027002544180023, −7.86609689007674990671503936944, −6.89057432670950296485638120161, −6.10975084183929983780369205919, −5.57892328762951052708961878118, −4.00010638002098044120664335315, −2.20129473282072452433620096169, −0.831035979955880272111811836334, 1.04720957746543300519499141383, 1.95427500840102451183935398832, 3.52307170944938480959799247413, 3.96763718444200866638952341471, 5.95259491233332255212686343963, 6.91803492472255449073404111519, 7.80891086098901090105421780081, 8.708896709653311252578991677665, 9.190547888805535951668800284471, 9.986903319936418125590750293969

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