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Properties

Label 2-891-11.10-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.344372109\)
\(L(\frac12)\) \(\approx\) \(1.344372109\)
\(L(1)\) 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

−10.29114147203813530209877639268, −9.786303477350711820419008740462, −8.590240338296545525717409753964, −7.69409713708088809799695670544, −6.90443336723620567388599182540, −5.83225783723782156121035770707, −5.48556159645197326045003082557, −3.89180589700010768822240525355, −2.56335030144389520857689117860, −1.85141390469645855415782041976, 1.85141390469645855415782041976, 2.56335030144389520857689117860, 3.89180589700010768822240525355, 5.48556159645197326045003082557, 5.83225783723782156121035770707, 6.90443336723620567388599182540, 7.69409713708088809799695670544, 8.590240338296545525717409753964, 9.786303477350711820419008740462, 10.29114147203813530209877639268

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