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Properties

Label 2-891-33.29-c1-0-0
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: $\chi_{891} (161, \cdot )$
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

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

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