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 $0$

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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