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Properties

Label 2-891-33.29-c1-0-21
Degree $2$
Conductor $891$
Sign $-0.621 - 0.783i$
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  + (2.17 + 1.58i)2-s + (1.62 + 4.99i)4-s + (0.850 + 1.17i)5-s + (1.37 − 0.446i)7-s + (−2.71 + 8.34i)8-s + 3.89i·10-s + (−0.371 − 3.29i)11-s + (−1.74 + 2.40i)13-s + (3.69 + 1.20i)14-s + (−10.6 + 7.71i)16-s + (3.35 − 2.44i)17-s + (−1.19 − 0.387i)19-s + (−4.47 + 6.15i)20-s + (4.40 − 7.77i)22-s + 5.80i·23-s + ⋯
L(s)  = 1  + (1.54 + 1.11i)2-s + (0.812 + 2.49i)4-s + (0.380 + 0.523i)5-s + (0.518 − 0.168i)7-s + (−0.958 + 2.95i)8-s + 1.23i·10-s + (−0.111 − 0.993i)11-s + (−0.485 + 0.667i)13-s + (0.988 + 0.321i)14-s + (−2.65 + 1.92i)16-s + (0.814 − 0.591i)17-s + (−0.273 − 0.0888i)19-s + (−0.999 + 1.37i)20-s + (0.940 − 1.65i)22-s + 1.21i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.621 - 0.783i$
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: \((2,\ 891,\ (\ :1/2),\ -0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72392 + 3.56928i\)
\(L(\frac12)\) \(\approx\) \(1.72392 + 3.56928i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.371 + 3.29i)T \)
good2 \( 1 + (-2.17 - 1.58i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-0.850 - 1.17i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.37 + 0.446i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.74 - 2.40i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.35 + 2.44i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.19 + 0.387i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.80iT - 23T^{2} \)
29 \( 1 + (-0.0300 - 0.0925i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.30 + 2.40i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.08 + 9.49i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.41 + 4.34i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.12iT - 43T^{2} \)
47 \( 1 + (-6.90 - 2.24i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.197 + 0.271i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.87 + 0.935i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.187 - 0.257i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.43T + 67T^{2} \)
71 \( 1 + (-5.38 - 7.41i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (6.75 - 2.19i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.88 - 8.10i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.82 + 3.50i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 5.11iT - 89T^{2} \)
97 \( 1 + (4.88 + 3.55i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.75710080049672432271165263605, −9.387222038446526825790136780467, −8.355376965584589172314982597568, −7.45529656123940576931251341092, −6.95017361042779761538460575366, −5.83390823940202560318359320395, −5.41355283177373259167062553277, −4.28726346091614207784732571845, −3.40676094178588992619996472439, −2.33952088582418613349474897400, 1.33278332345048808853396906070, 2.28006040410921672248856409156, 3.37501582214052932526297313586, 4.57774487443771398798030081656, 5.05970101721967693544954735856, 5.87156680895578816471062366097, 6.91613373762033965576144891779, 8.178429600391835491314662467925, 9.408192660527166028582819386798, 10.21802173731051654103211675786

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