Properties

Label 2-6864-1.1-c1-0-11
Degree $2$
Conductor $6864$
Sign $1$
Analytic cond. $54.8093$
Root an. cond. $7.40333$
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  − 3-s − 0.447·5-s − 3.31·7-s + 9-s − 11-s + 13-s + 0.447·15-s + 4.86·17-s − 3.83·19-s + 3.31·21-s + 5.31·23-s − 4.79·25-s − 27-s + 8.83·29-s − 1.93·31-s + 33-s + 1.48·35-s − 6.63·37-s − 39-s − 1.31·41-s − 7.24·43-s − 0.447·45-s − 4.48·47-s + 3.99·49-s − 4.86·51-s − 11.1·53-s + 0.447·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.200·5-s − 1.25·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s + 0.115·15-s + 1.18·17-s − 0.879·19-s + 0.723·21-s + 1.10·23-s − 0.959·25-s − 0.192·27-s + 1.64·29-s − 0.347·31-s + 0.174·33-s + 0.250·35-s − 1.09·37-s − 0.160·39-s − 0.205·41-s − 1.10·43-s − 0.0667·45-s − 0.653·47-s + 0.571·49-s − 0.681·51-s − 1.52·53-s + 0.0603·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6864\)    =    \(2^{4} \cdot 3 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(54.8093\)
Root analytic conductor: \(7.40333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9093694882\)
\(L(\frac12)\) \(\approx\) \(0.9093694882\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 + 0.447T + 5T^{2} \)
7 \( 1 + 3.31T + 7T^{2} \)
17 \( 1 - 4.86T + 17T^{2} \)
19 \( 1 + 3.83T + 19T^{2} \)
23 \( 1 - 5.31T + 23T^{2} \)
29 \( 1 - 8.83T + 29T^{2} \)
31 \( 1 + 1.93T + 31T^{2} \)
37 \( 1 + 6.63T + 37T^{2} \)
41 \( 1 + 1.31T + 41T^{2} \)
43 \( 1 + 7.24T + 43T^{2} \)
47 \( 1 + 4.48T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 - 5.63T + 59T^{2} \)
61 \( 1 - 0.380T + 61T^{2} \)
67 \( 1 + 2.44T + 67T^{2} \)
71 \( 1 - 7.25T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 + 0.895T + 83T^{2} \)
89 \( 1 + 9.46T + 89T^{2} \)
97 \( 1 + 4.22T + 97T^{2} \)
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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

−8.035382898354909423461485555008, −6.96322656523375113757840625474, −6.66969407524774172106117789725, −5.88283111321071248543277405249, −5.23130587813916526797584781854, −4.40830513081362643760149604752, −3.44899099254231110889819071055, −2.98691586252601523140170014228, −1.68585932633609670389577207378, −0.49991290604070523834266395960, 0.49991290604070523834266395960, 1.68585932633609670389577207378, 2.98691586252601523140170014228, 3.44899099254231110889819071055, 4.40830513081362643760149604752, 5.23130587813916526797584781854, 5.88283111321071248543277405249, 6.66969407524774172106117789725, 6.96322656523375113757840625474, 8.035382898354909423461485555008

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