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Properties

Label 2-891-11.3-c1-0-43
Degree $2$
Conductor $891$
Sign $-0.995 - 0.0913i$
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  + (2.11 − 1.53i)2-s + (1.5 − 4.61i)4-s + (−2.61 − 1.90i)5-s + (−0.309 + 0.951i)7-s + (−2.30 − 7.10i)8-s − 8.47·10-s + (−0.809 − 3.21i)11-s + (−2 + 1.45i)13-s + (0.809 + 2.48i)14-s + (−7.97 − 5.79i)16-s + (1.5 + 1.08i)17-s + (0.309 + 0.951i)19-s + (−12.7 + 9.23i)20-s + (−6.66 − 5.56i)22-s + 2.38·23-s + ⋯
L(s)  = 1  + (1.49 − 1.08i)2-s + (0.750 − 2.30i)4-s + (−1.17 − 0.850i)5-s + (−0.116 + 0.359i)7-s + (−0.816 − 2.51i)8-s − 2.67·10-s + (−0.243 − 0.969i)11-s + (−0.554 + 0.403i)13-s + (0.216 + 0.665i)14-s + (−1.99 − 1.44i)16-s + (0.363 + 0.264i)17-s + (0.0708 + 0.218i)19-s + (−2.84 + 2.06i)20-s + (−1.42 − 1.18i)22-s + 0.496·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.995 - 0.0913i$
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: \((2,\ 891,\ (\ :1/2),\ -0.995 - 0.0913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.112036 + 2.44665i\)
\(L(\frac12)\) \(\approx\) \(0.112036 + 2.44665i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 3.21i)T \)
good2 \( 1 + (-2.11 + 1.53i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (2.61 + 1.90i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.309 - 0.951i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2 - 1.45i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.5 - 1.08i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.38T + 23T^{2} \)
29 \( 1 + (-0.572 + 1.76i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.30 + 0.951i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.88 + 5.79i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.54 + 10.9i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.85T + 43T^{2} \)
47 \( 1 + (1.83 + 5.65i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.73 + 4.16i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.190 - 0.587i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.35 + 2.43i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 6T + 67T^{2} \)
71 \( 1 + (-11.3 - 8.28i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.11 - 6.51i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.73 + 5.62i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-11.5 - 8.36i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (4.85 - 3.52i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.00756136347978503381444230573, −8.963494547706448669014503173861, −8.083550085987528502808109602866, −6.86861989254539340709474062377, −5.66283861682383227841124687221, −5.08496041897666157795569777859, −4.07187702317889266878213895630, −3.47179604253122515555837601275, −2.26781808758544159168998947588, −0.71248925022986609927474814306, 2.79036428106843959681131277779, 3.46520017265420928644145779666, 4.50616976434005615224285706875, 5.08469084509030989933315259519, 6.41401066780639407770903699851, 7.06061603740376869479139717241, 7.60604949451991617988826767995, 8.243120632728134287386082097436, 9.787049511574524761636854477329, 10.82405851610382698080383430724

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