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Properties

Label 2-891-11.10-c0-0-0
Degree $2$
Conductor $891$
Sign $1$
Analytic cond. $0.444666$
Root an. cond. $0.666833$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 5-s + 11-s + 16-s − 20-s + 2·23-s − 31-s − 37-s + 44-s − 47-s + 49-s − 53-s − 55-s − 59-s + 64-s − 67-s − 71-s − 80-s + 2·89-s + 2·92-s − 97-s − 103-s − 113-s − 2·115-s + ⋯
L(s)  = 1  + 4-s − 5-s + 11-s + 16-s − 20-s + 2·23-s − 31-s − 37-s + 44-s − 47-s + 49-s − 53-s − 55-s − 59-s + 64-s − 67-s − 71-s − 80-s + 2·89-s + 2·92-s − 97-s − 103-s − 113-s − 2·115-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.444666\)
Root analytic conductor: \(0.666833\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{891} (406, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.119384270\)
\(L(\frac12)\) \(\approx\) \(1.119384270\)
\(L(1)\) 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 \( 1 - T \)
good2 \( ( 1 - T )( 1 + T ) \)
5 \( 1 + T + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )^{2} \)
97 \( 1 + T + T^{2} \)
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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

−10.63573883412275318061985997836, −9.409721067535093932371512951938, −8.620487409826410484883836101317, −7.57024769225928312000223485123, −7.04840010366187114546216085664, −6.20428371488628605414703670018, −5.01899102007676319241703606728, −3.82105499787674479863267802941, −3.02265685485842577322213876178, −1.49992246792186455847727789153, 1.49992246792186455847727789153, 3.02265685485842577322213876178, 3.82105499787674479863267802941, 5.01899102007676319241703606728, 6.20428371488628605414703670018, 7.04840010366187114546216085664, 7.57024769225928312000223485123, 8.620487409826410484883836101317, 9.409721067535093932371512951938, 10.63573883412275318061985997836

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