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Properties

Label 2-891-11.3-c1-0-7
Degree $2$
Conductor $891$
Sign $-0.999 + 0.0252i$
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  + (−0.537 + 0.390i)2-s + (−0.481 + 1.48i)4-s + (0.903 + 0.656i)5-s + (−1.20 + 3.71i)7-s + (−0.730 − 2.24i)8-s − 0.741·10-s + (1.17 + 3.10i)11-s + (−2.16 + 1.57i)13-s + (−0.801 − 2.46i)14-s + (−1.25 − 0.909i)16-s + (4.48 + 3.26i)17-s + (−1.58 − 4.88i)19-s + (−1.40 + 1.02i)20-s + (−1.84 − 1.20i)22-s − 2.11·23-s + ⋯
L(s)  = 1  + (−0.380 + 0.276i)2-s + (−0.240 + 0.741i)4-s + (0.403 + 0.293i)5-s + (−0.456 + 1.40i)7-s + (−0.258 − 0.794i)8-s − 0.234·10-s + (0.355 + 0.934i)11-s + (−0.600 + 0.436i)13-s + (−0.214 − 0.659i)14-s + (−0.312 − 0.227i)16-s + (1.08 + 0.790i)17-s + (−0.363 − 1.11i)19-s + (−0.314 + 0.228i)20-s + (−0.393 − 0.257i)22-s − 0.440·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.999 + 0.0252i$
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: \((2,\ 891,\ (\ :1/2),\ -0.999 + 0.0252i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0105923 - 0.839732i\)
\(L(\frac12)\) \(\approx\) \(0.0105923 - 0.839732i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.17 - 3.10i)T \)
good2 \( 1 + (0.537 - 0.390i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-0.903 - 0.656i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.20 - 3.71i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.16 - 1.57i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.48 - 3.26i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.58 + 4.88i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.11T + 23T^{2} \)
29 \( 1 + (0.334 - 1.03i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.348 + 0.252i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.69 - 8.27i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.902 - 2.77i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.23T + 43T^{2} \)
47 \( 1 + (3.96 + 12.2i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.56 + 3.31i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.734 + 2.25i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.30 - 2.40i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.75T + 67T^{2} \)
71 \( 1 + (-5.11 - 3.71i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.60 - 11.1i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.65 - 2.65i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.389 - 0.283i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 + (6.48 - 4.71i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.01710185743282912815381411321, −9.762949137600461926479760902381, −8.781125938069610698089244470242, −8.231522208277325703548918514828, −7.03774156221530688772673080741, −6.49311434785654853202686441741, −5.39795477135129715531035646073, −4.28156104164438816164471208815, −3.05092194353897268217206027591, −2.10540472026488481882898425302, 0.46275315951784902422942900184, 1.53556641209884429987447613982, 3.17761831448396582041464729047, 4.24734550213935252086596695725, 5.46497593583626164717513174386, 6.03885196151899997433211672333, 7.25499135091433946852239866309, 8.050321101540348295015947009640, 9.167813632825811163595388521698, 9.795728371388316739502120924027

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