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Properties

Label 2-891-9.7-c1-0-33
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: $\chi_{891} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.367326 - 2.08321i\)
\(L(\frac12)\) \(\approx\) \(0.367326 - 2.08321i\)
\(L(\frac{3}{2})\) 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

−10.08743329892183934504542722219, −9.200389980300682870852647434820, −7.914510205276178384785258175379, −7.44131179812502982966235398717, −5.98104958018836848049172146501, −4.80420337545370249784913092303, −4.36835539561987769244284227735, −3.23120566598701975081249975262, −2.19989841012409930974517391371, −0.821839682517520122027716419683, 2.01485887818423098635907290802, 3.45688696280049179578436652359, 4.61347898967510088240709179001, 5.25704252722515264182910305830, 6.25024784498824974826541263962, 6.96287025037577745235969306248, 7.67774524044085329807655639786, 8.615933931568244902023499452891, 9.365701588977284945576055223209, 10.66295659293859806497800795290

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