Properties

Label 2-867-1.1-c1-0-21
Degree $2$
Conductor $867$
Sign $1$
Analytic cond. $6.92302$
Root an. cond. $2.63116$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56·2-s + 3-s + 0.438·4-s + 0.561·5-s + 1.56·6-s − 2.43·8-s + 9-s + 0.876·10-s + 2.56·11-s + 0.438·12-s + 4.56·13-s + 0.561·15-s − 4.68·16-s + 1.56·18-s + 7.68·19-s + 0.246·20-s + 4·22-s + 6.56·23-s − 2.43·24-s − 4.68·25-s + 7.12·26-s + 27-s − 8.24·29-s + 0.876·30-s + 5.12·31-s − 2.43·32-s + 2.56·33-s + ⋯
L(s)  = 1  + 1.10·2-s + 0.577·3-s + 0.219·4-s + 0.251·5-s + 0.637·6-s − 0.862·8-s + 0.333·9-s + 0.277·10-s + 0.772·11-s + 0.126·12-s + 1.26·13-s + 0.144·15-s − 1.17·16-s + 0.368·18-s + 1.76·19-s + 0.0550·20-s + 0.852·22-s + 1.36·23-s − 0.497·24-s − 0.936·25-s + 1.39·26-s + 0.192·27-s − 1.53·29-s + 0.160·30-s + 0.920·31-s − 0.431·32-s + 0.445·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.346552571\)
\(L(\frac12)\) \(\approx\) \(3.346552571\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
17 \( 1 \)
good2 \( 1 - 1.56T + 2T^{2} \)
5 \( 1 - 0.561T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
19 \( 1 - 7.68T + 19T^{2} \)
23 \( 1 - 6.56T + 23T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 - 5.12T + 31T^{2} \)
37 \( 1 + 3.12T + 37T^{2} \)
41 \( 1 + 0.561T + 41T^{2} \)
43 \( 1 + 7.68T + 43T^{2} \)
47 \( 1 + 2.87T + 47T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 + 1.12T + 59T^{2} \)
61 \( 1 + 0.876T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 + 4.24T + 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 - 7.12T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−10.00375561172223746959390243295, −9.242013867079147294595552611063, −8.645072637455036397263998565000, −7.48739150606086712838347985580, −6.48277922844872059298946487837, −5.66787098945815039306128222831, −4.75018432724652231899341697672, −3.63495108992766008145248931444, −3.14573716219350256540938720399, −1.48567602305201078849771570389, 1.48567602305201078849771570389, 3.14573716219350256540938720399, 3.63495108992766008145248931444, 4.75018432724652231899341697672, 5.66787098945815039306128222831, 6.48277922844872059298946487837, 7.48739150606086712838347985580, 8.645072637455036397263998565000, 9.242013867079147294595552611063, 10.00375561172223746959390243295

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