Properties

Label 2-6864-1.1-c1-0-69
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 + 3.20·5-s + 1.53·7-s + 9-s − 11-s − 13-s + 3.20·15-s + 2.73·17-s + 7.78·19-s + 1.53·21-s + 6.88·23-s + 5.28·25-s + 27-s + 1.22·29-s + 8.11·31-s − 33-s + 4.90·35-s − 6.41·37-s − 39-s + 3.53·41-s − 3.98·43-s + 3.20·45-s + 2.93·47-s − 4.65·49-s + 2.73·51-s − 10.2·53-s − 3.20·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.43·5-s + 0.578·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s + 0.827·15-s + 0.663·17-s + 1.78·19-s + 0.333·21-s + 1.43·23-s + 1.05·25-s + 0.192·27-s + 0.228·29-s + 1.45·31-s − 0.174·33-s + 0.829·35-s − 1.05·37-s − 0.160·39-s + 0.551·41-s − 0.607·43-s + 0.478·45-s + 0.428·47-s − 0.665·49-s + 0.383·51-s − 1.40·53-s − 0.432·55-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: $\chi_{6864} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.168116226\)
\(L(\frac12)\) \(\approx\) \(4.168116226\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.20T + 5T^{2} \)
7 \( 1 - 1.53T + 7T^{2} \)
17 \( 1 - 2.73T + 17T^{2} \)
19 \( 1 - 7.78T + 19T^{2} \)
23 \( 1 - 6.88T + 23T^{2} \)
29 \( 1 - 1.22T + 29T^{2} \)
31 \( 1 - 8.11T + 31T^{2} \)
37 \( 1 + 6.41T + 37T^{2} \)
41 \( 1 - 3.53T + 41T^{2} \)
43 \( 1 + 3.98T + 43T^{2} \)
47 \( 1 - 2.93T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 + 6.30T + 61T^{2} \)
67 \( 1 - 5.80T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 + 15.4T + 73T^{2} \)
79 \( 1 - 1.98T + 79T^{2} \)
83 \( 1 + 5.16T + 83T^{2} \)
89 \( 1 + 4.11T + 89T^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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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

−7.85943809152726599834738967121, −7.41640733116348374664292778214, −6.55000047563419362637934970541, −5.78872801896530839714048810997, −5.08275836219410314073456125978, −4.64820317927332447274639330747, −3.15262503568123356316598015924, −2.88524984795657189862372587537, −1.73115356986566539217591542723, −1.14432576572104660776279558679, 1.14432576572104660776279558679, 1.73115356986566539217591542723, 2.88524984795657189862372587537, 3.15262503568123356316598015924, 4.64820317927332447274639330747, 5.08275836219410314073456125978, 5.78872801896530839714048810997, 6.55000047563419362637934970541, 7.41640733116348374664292778214, 7.85943809152726599834738967121

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