Properties

Label 2-6864-1.1-c1-0-25
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 − 3.34·5-s + 2.51·7-s + 9-s − 11-s + 13-s + 3.34·15-s + 7.59·17-s + 1.48·19-s − 2.51·21-s + 1.55·23-s + 6.21·25-s − 27-s + 1.86·29-s + 1.61·31-s + 33-s − 8.43·35-s + 6.43·37-s − 39-s − 10.9·41-s + 5.59·43-s − 3.34·45-s − 0.697·47-s − 0.663·49-s − 7.59·51-s − 9.73·53-s + 3.34·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.49·5-s + 0.951·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s + 0.864·15-s + 1.84·17-s + 0.340·19-s − 0.549·21-s + 0.323·23-s + 1.24·25-s − 0.192·27-s + 0.346·29-s + 0.290·31-s + 0.174·33-s − 1.42·35-s + 1.05·37-s − 0.160·39-s − 1.70·41-s + 0.853·43-s − 0.499·45-s − 0.101·47-s − 0.0947·49-s − 1.06·51-s − 1.33·53-s + 0.451·55-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

\(L(1)\) \(\approx\) \(1.366665906\)
\(L(\frac12)\) \(\approx\) \(1.366665906\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.34T + 5T^{2} \)
7 \( 1 - 2.51T + 7T^{2} \)
17 \( 1 - 7.59T + 17T^{2} \)
19 \( 1 - 1.48T + 19T^{2} \)
23 \( 1 - 1.55T + 23T^{2} \)
29 \( 1 - 1.86T + 29T^{2} \)
31 \( 1 - 1.61T + 31T^{2} \)
37 \( 1 - 6.43T + 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 - 5.59T + 43T^{2} \)
47 \( 1 + 0.697T + 47T^{2} \)
53 \( 1 + 9.73T + 53T^{2} \)
59 \( 1 - 4.76T + 59T^{2} \)
61 \( 1 + 1.66T + 61T^{2} \)
67 \( 1 + 4.81T + 67T^{2} \)
71 \( 1 - 0.965T + 71T^{2} \)
73 \( 1 + 3.21T + 73T^{2} \)
79 \( 1 - 8.83T + 79T^{2} \)
83 \( 1 - 0.767T + 83T^{2} \)
89 \( 1 - 3.41T + 89T^{2} \)
97 \( 1 + 6.96T + 97T^{2} \)
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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

−7.954265363473848770720260003911, −7.47273201296015151243799636154, −6.66234422622519743953474196914, −5.71935454302311262707885985748, −5.03804923039050650054704908871, −4.47413620749754394392723396472, −3.63516285062417893401090891861, −2.96989125042364779647030262805, −1.50738121371879159353853034489, −0.66726691544710570416462432806, 0.66726691544710570416462432806, 1.50738121371879159353853034489, 2.96989125042364779647030262805, 3.63516285062417893401090891861, 4.47413620749754394392723396472, 5.03804923039050650054704908871, 5.71935454302311262707885985748, 6.66234422622519743953474196914, 7.47273201296015151243799636154, 7.954265363473848770720260003911

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