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Properties

Label 2-891-11.3-c1-0-1
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

\(L(1)\) \(\approx\) \(0.00313446 - 0.248491i\)
\(L(\frac12)\) \(\approx\) \(0.00313446 - 0.248491i\)
\(L(\frac{3}{2})\) 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

−10.81674438032469175153339734650, −9.385035461703761197145522014911, −8.844619536392860011090775066152, −8.270658898310627675558199566861, −7.16429977433340249764266014758, −6.15598538229805280981916041099, −5.02543262283195631696436358317, −4.39213979010353410280182854374, −2.96181729076216591248220582151, −2.49944841795214544982682157315, 0.098975345366517127934781576575, 1.80777747624913365497027391451, 3.59460254458945808487480803691, 4.27951288744508518598733723685, 5.19740077419748683883275310594, 6.33600594442951836022897862489, 7.09850613312626439357945598662, 7.65902255220121062576356415579, 8.954919326748970610724281470052, 10.05498779233529035518455525552

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