Properties

Label 2-6864-1.1-c1-0-1
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3.29·5-s − 3.59·7-s + 9-s − 11-s + 13-s − 3.29·15-s − 8.12·17-s − 3.59·19-s − 3.59·21-s + 3.74·23-s + 5.83·25-s + 27-s + 3.69·29-s − 5.29·31-s − 33-s + 11.8·35-s − 9.82·37-s + 39-s + 8.08·41-s − 2.94·43-s − 3.29·45-s − 8.73·47-s + 5.90·49-s − 8.12·51-s − 8.58·53-s + 3.29·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.47·5-s − 1.35·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s − 0.849·15-s − 1.97·17-s − 0.824·19-s − 0.784·21-s + 0.781·23-s + 1.16·25-s + 0.192·27-s + 0.686·29-s − 0.950·31-s − 0.174·33-s + 1.99·35-s − 1.61·37-s + 0.160·39-s + 1.26·41-s − 0.448·43-s − 0.490·45-s − 1.27·47-s + 0.844·49-s − 1.13·51-s − 1.17·53-s + 0.443·55-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

\(L(1)\) \(\approx\) \(0.5434139545\)
\(L(\frac12)\) \(\approx\) \(0.5434139545\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.29T + 5T^{2} \)
7 \( 1 + 3.59T + 7T^{2} \)
17 \( 1 + 8.12T + 17T^{2} \)
19 \( 1 + 3.59T + 19T^{2} \)
23 \( 1 - 3.74T + 23T^{2} \)
29 \( 1 - 3.69T + 29T^{2} \)
31 \( 1 + 5.29T + 31T^{2} \)
37 \( 1 + 9.82T + 37T^{2} \)
41 \( 1 - 8.08T + 41T^{2} \)
43 \( 1 + 2.94T + 43T^{2} \)
47 \( 1 + 8.73T + 47T^{2} \)
53 \( 1 + 8.58T + 53T^{2} \)
59 \( 1 + 6.58T + 59T^{2} \)
61 \( 1 - 7.24T + 61T^{2} \)
67 \( 1 - 4.38T + 67T^{2} \)
71 \( 1 + 9.33T + 71T^{2} \)
73 \( 1 + 2.99T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 - 4.09T + 83T^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.044499349127200563832343424624, −7.14791885447494715854027841698, −6.80371916382468402327964353926, −6.09760941157462448614802996316, −4.80733807640413162702697218669, −4.26816339361242819182176339197, −3.46886979258412074696017677891, −3.01165092238520233557687253360, −1.97460818253951965966818265759, −0.34307159467423699625774946546, 0.34307159467423699625774946546, 1.97460818253951965966818265759, 3.01165092238520233557687253360, 3.46886979258412074696017677891, 4.26816339361242819182176339197, 4.80733807640413162702697218669, 6.09760941157462448614802996316, 6.80371916382468402327964353926, 7.14791885447494715854027841698, 8.044499349127200563832343424624

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