Properties

Label 2-6864-1.1-c1-0-13
Degree $2$
Conductor $6864$
Sign $1$
Analytic cond. $54.8093$
Root an. cond. $7.40333$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s − 4·7-s + 9-s − 11-s − 13-s − 2·15-s − 8·17-s + 6·19-s + 4·21-s + 6·23-s − 25-s − 27-s − 2·29-s + 2·31-s + 33-s − 8·35-s + 4·37-s + 39-s + 2·41-s − 4·43-s + 2·45-s − 4·47-s + 9·49-s + 8·51-s − 8·53-s − 2·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.516·15-s − 1.94·17-s + 1.37·19-s + 0.872·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.359·31-s + 0.174·33-s − 1.35·35-s + 0.657·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s − 0.583·47-s + 9/7·49-s + 1.12·51-s − 1.09·53-s − 0.269·55-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.106464907\)
\(L(\frac12)\) \(\approx\) \(1.106464907\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.77449283407253786009460886713, −7.06563821912503711347301271655, −6.37527899037507766078006035833, −6.10020985198044617350989038822, −5.14909446151884675735052829793, −4.59172296811147478255924579412, −3.42153359247955952503422970484, −2.77339832615760976506042506736, −1.83546111330411908461338448749, −0.53483098955708202235056274838, 0.53483098955708202235056274838, 1.83546111330411908461338448749, 2.77339832615760976506042506736, 3.42153359247955952503422970484, 4.59172296811147478255924579412, 5.14909446151884675735052829793, 6.10020985198044617350989038822, 6.37527899037507766078006035833, 7.06563821912503711347301271655, 7.77449283407253786009460886713

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