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Properties

Label 2-891-11.3-c1-0-13
Degree $2$
Conductor $891$
Sign $0.698 - 0.715i$
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  + (1.48 − 1.07i)2-s + (0.421 − 1.29i)4-s + (0.0729 + 0.0530i)5-s + (−1.45 + 4.48i)7-s + (0.360 + 1.11i)8-s + 0.165·10-s + (−3.00 − 1.39i)11-s + (−2.65 + 1.92i)13-s + (2.67 + 8.21i)14-s + (3.93 + 2.86i)16-s + (3.80 + 2.76i)17-s + (1.79 + 5.53i)19-s + (0.0995 − 0.0723i)20-s + (−5.96 + 1.16i)22-s − 2.20·23-s + ⋯
L(s)  = 1  + (1.04 − 0.762i)2-s + (0.210 − 0.648i)4-s + (0.0326 + 0.0237i)5-s + (−0.550 + 1.69i)7-s + (0.127 + 0.392i)8-s + 0.0523·10-s + (−0.906 − 0.421i)11-s + (−0.735 + 0.534i)13-s + (0.713 + 2.19i)14-s + (0.984 + 0.715i)16-s + (0.924 + 0.671i)17-s + (0.412 + 1.27i)19-s + (0.0222 − 0.0161i)20-s + (−1.27 + 0.248i)22-s − 0.458·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.698 - 0.715i$
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: \((2,\ 891,\ (\ :1/2),\ 0.698 - 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00257 + 0.843580i\)
\(L(\frac12)\) \(\approx\) \(2.00257 + 0.843580i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.00 + 1.39i)T \)
good2 \( 1 + (-1.48 + 1.07i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-0.0729 - 0.0530i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.45 - 4.48i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.65 - 1.92i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.80 - 2.76i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.79 - 5.53i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.20T + 23T^{2} \)
29 \( 1 + (-1.57 + 4.85i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.95 + 2.87i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.41 - 4.34i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.32 - 7.14i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 5.79T + 43T^{2} \)
47 \( 1 + (0.290 + 0.893i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-9.93 + 7.21i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.195 + 0.602i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.944 - 0.686i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.12T + 67T^{2} \)
71 \( 1 + (-6.15 - 4.47i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.24 + 9.98i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.27 - 5.28i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-11.4 - 8.33i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 2.58T + 89T^{2} \)
97 \( 1 + (-4.00 + 2.91i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.12102116439718185163999036927, −9.820072462878761085733681494700, −8.337360247025792520191843321808, −8.039062794233897692036551266131, −6.26846737539008145249290779449, −5.70705637130474923475037826068, −4.93209784397747121603002043669, −3.71875131675853201981242488111, −2.77248711273341353606849046854, −2.06624439696551477434023977932, 0.72886308652330044803633918612, 2.94921119851478350248338868027, 3.83204835558636786204875861402, 4.90245408072371147017065330791, 5.38001688536976330923667487950, 6.70457356501171026564431877291, 7.33376017061547672281689872214, 7.68686340239109680975688756988, 9.338454352006755947392253180220, 10.19637202007502149669527355479

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