Properties

Label 2-6864-1.1-c1-0-2
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 − 3.73·5-s − 0.404·7-s + 9-s − 11-s + 13-s + 3.73·15-s − 1.32·17-s − 0.895·19-s + 0.404·21-s + 2.40·23-s + 8.92·25-s − 27-s − 8.14·29-s − 5.23·31-s + 33-s + 1.50·35-s − 0.808·37-s − 39-s + 1.59·41-s − 7.64·43-s − 3.73·45-s + 6.32·47-s − 6.83·49-s + 1.32·51-s + 5.51·53-s + 3.73·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.66·5-s − 0.152·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s + 0.963·15-s − 0.321·17-s − 0.205·19-s + 0.0882·21-s + 0.501·23-s + 1.78·25-s − 0.192·27-s − 1.51·29-s − 0.941·31-s + 0.174·33-s + 0.254·35-s − 0.132·37-s − 0.160·39-s + 0.249·41-s − 1.16·43-s − 0.556·45-s + 0.923·47-s − 0.976·49-s + 0.185·51-s + 0.758·53-s + 0.503·55-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

\(L(1)\) \(\approx\) \(0.4570595534\)
\(L(\frac12)\) \(\approx\) \(0.4570595534\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.73T + 5T^{2} \)
7 \( 1 + 0.404T + 7T^{2} \)
17 \( 1 + 1.32T + 17T^{2} \)
19 \( 1 + 0.895T + 19T^{2} \)
23 \( 1 - 2.40T + 23T^{2} \)
29 \( 1 + 8.14T + 29T^{2} \)
31 \( 1 + 5.23T + 31T^{2} \)
37 \( 1 + 0.808T + 37T^{2} \)
41 \( 1 - 1.59T + 41T^{2} \)
43 \( 1 + 7.64T + 43T^{2} \)
47 \( 1 - 6.32T + 47T^{2} \)
53 \( 1 - 5.51T + 53T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 - 6.97T + 61T^{2} \)
67 \( 1 + 5.73T + 67T^{2} \)
71 \( 1 + 15.9T + 71T^{2} \)
73 \( 1 + 1.92T + 73T^{2} \)
79 \( 1 - 2.40T + 79T^{2} \)
83 \( 1 + 7.46T + 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−7.72185965013121304755333036187, −7.39191605417648692965790939041, −6.69786853869800457853480737998, −5.82406941178486761749977633808, −5.06175522420810728079885182099, −4.29889896091032569546861571460, −3.71710814530913938887336401668, −2.96376126291767695773861200308, −1.64253520715635596788854387636, −0.35521481485018965449200325244, 0.35521481485018965449200325244, 1.64253520715635596788854387636, 2.96376126291767695773861200308, 3.71710814530913938887336401668, 4.29889896091032569546861571460, 5.06175522420810728079885182099, 5.82406941178486761749977633808, 6.69786853869800457853480737998, 7.39191605417648692965790939041, 7.72185965013121304755333036187

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