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Properties

Label 2-891-9.4-c1-0-8
Degree $2$
Conductor $891$
Sign $0.939 + 0.342i$
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 − 1.73i)2-s + (−0.999 + 1.73i)4-s + (0.5 − 0.866i)5-s + (1 + 1.73i)7-s − 1.99·10-s + (0.5 + 0.866i)11-s + (−2 + 3.46i)13-s + (1.99 − 3.46i)14-s + (1.99 + 3.46i)16-s + 2·17-s + (1 + 1.73i)20-s + (0.999 − 1.73i)22-s + (−0.5 + 0.866i)23-s + (2 + 3.46i)25-s + 7.99·26-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.499 + 0.866i)4-s + (0.223 − 0.387i)5-s + (0.377 + 0.654i)7-s − 0.632·10-s + (0.150 + 0.261i)11-s + (−0.554 + 0.960i)13-s + (0.534 − 0.925i)14-s + (0.499 + 0.866i)16-s + 0.485·17-s + (0.223 + 0.387i)20-s + (0.213 − 0.369i)22-s + (−0.104 + 0.180i)23-s + (0.400 + 0.692i)25-s + 1.56·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.957769 - 0.168880i\)
\(L(\frac12)\) \(\approx\) \(0.957769 - 0.168880i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (1 + 1.73i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (4 - 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (-3.5 - 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.925569538625103786503038661764, −9.388539370475559419808807313705, −8.788315572261211621855054424566, −7.889915417111754019794747330173, −6.71699172988457753770502936521, −5.57013222367304179745119260844, −4.60707662421296056449984401712, −3.32673679608440025780294036832, −2.20628019169164723982645620548, −1.33032215562267134777673100485, 0.64394575992025308569407129602, 2.60710808575992414991025440165, 3.94252680510148667732513130689, 5.31121536616716968733234337672, 5.96604595338368012249418257466, 7.03381138056625713353507603218, 7.52369573823768515626195108496, 8.311485914046732258942545650122, 9.114729009673412788752942672141, 10.14157832086397313419143910163

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