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Properties

Label 2-891-33.8-c1-0-42
Degree $2$
Conductor $891$
Sign $-0.791 - 0.610i$
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  + (1.46 − 1.06i)2-s + (0.399 − 1.22i)4-s + (−0.706 + 0.972i)5-s + (−4.60 − 1.49i)7-s + (0.396 + 1.22i)8-s + 2.18i·10-s + (−3.30 − 0.324i)11-s + (−1.79 − 2.47i)13-s + (−8.35 + 2.71i)14-s + (3.97 + 2.88i)16-s + (−1.80 − 1.31i)17-s + (−4.34 + 1.41i)19-s + (0.912 + 1.25i)20-s + (−5.19 + 3.04i)22-s − 1.98i·23-s + ⋯
L(s)  = 1  + (1.03 − 0.754i)2-s + (0.199 − 0.614i)4-s + (−0.315 + 0.434i)5-s + (−1.74 − 0.565i)7-s + (0.140 + 0.431i)8-s + 0.689i·10-s + (−0.995 − 0.0979i)11-s + (−0.498 − 0.686i)13-s + (−2.23 + 0.725i)14-s + (0.993 + 0.722i)16-s + (−0.438 − 0.318i)17-s + (−0.996 + 0.323i)19-s + (0.204 + 0.280i)20-s + (−1.10 + 0.648i)22-s − 0.414i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.791 - 0.610i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ -0.791 - 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0244796 + 0.0718309i\)
\(L(\frac12)\) \(\approx\) \(0.0244796 + 0.0718309i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.30 + 0.324i)T \)
good2 \( 1 + (-1.46 + 1.06i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (0.706 - 0.972i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (4.60 + 1.49i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.79 + 2.47i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.80 + 1.31i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.34 - 1.41i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.98iT - 23T^{2} \)
29 \( 1 + (-0.524 + 1.61i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.29 - 3.12i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.344 + 1.06i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.28 + 10.1i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.79iT - 43T^{2} \)
47 \( 1 + (3.10 - 1.00i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.749 - 1.03i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.0300 + 0.00976i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.697 - 0.960i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.10T + 67T^{2} \)
71 \( 1 + (-2.26 + 3.11i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-6.69 - 2.17i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.24 + 3.09i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.88 - 4.27i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 + (11.7 - 8.52i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.08192818104176457574253035983, −8.893619848676493925683656969760, −7.74889307335484388336454932107, −6.93266709190791092604134490238, −5.96696606730002263281180043770, −4.99422667522692489327476042251, −3.87085516657811759011242723811, −3.18908100143925649333692142937, −2.42183566845850938026633404581, −0.02330032319605712049353706441, 2.49508993504237604635080377308, 3.60786910773166350335342704457, 4.55267603517587755395138890269, 5.40293588914935841222571601361, 6.37310740272031465569545389212, 6.79389581567711609691005433115, 7.897764847095329485283685531852, 8.956035252178131654578997974763, 9.731407424288742617684117155155, 10.48594695559623937987217891263

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