Properties

Label 2-6864-1.1-c1-0-19
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 + 0.699·5-s − 3.79·7-s + 9-s − 11-s − 13-s + 0.699·15-s − 5.09·17-s + 0.538·19-s − 3.79·21-s + 7.18·23-s − 4.51·25-s + 27-s − 9.14·29-s − 1.95·31-s − 33-s − 2.65·35-s − 1.39·37-s − 39-s − 1.79·41-s + 10.8·43-s + 0.699·45-s + 13.5·47-s + 7.37·49-s − 5.09·51-s + 9.90·53-s − 0.699·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.312·5-s − 1.43·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s + 0.180·15-s − 1.23·17-s + 0.123·19-s − 0.827·21-s + 1.49·23-s − 0.902·25-s + 0.192·27-s − 1.69·29-s − 0.350·31-s − 0.174·33-s − 0.448·35-s − 0.229·37-s − 0.160·39-s − 0.279·41-s + 1.65·43-s + 0.104·45-s + 1.98·47-s + 1.05·49-s − 0.712·51-s + 1.36·53-s − 0.0942·55-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

\(L(1)\) \(\approx\) \(1.727956506\)
\(L(\frac12)\) \(\approx\) \(1.727956506\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.699T + 5T^{2} \)
7 \( 1 + 3.79T + 7T^{2} \)
17 \( 1 + 5.09T + 17T^{2} \)
19 \( 1 - 0.538T + 19T^{2} \)
23 \( 1 - 7.18T + 23T^{2} \)
29 \( 1 + 9.14T + 29T^{2} \)
31 \( 1 + 1.95T + 31T^{2} \)
37 \( 1 + 1.39T + 37T^{2} \)
41 \( 1 + 1.79T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 - 13.5T + 47T^{2} \)
53 \( 1 - 9.90T + 53T^{2} \)
59 \( 1 + 5.72T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 - 9.90T + 71T^{2} \)
73 \( 1 - 3.83T + 73T^{2} \)
79 \( 1 + 1.37T + 79T^{2} \)
83 \( 1 - 4.32T + 83T^{2} \)
89 \( 1 - 5.95T + 89T^{2} \)
97 \( 1 - 1.95T + 97T^{2} \)
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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

−7.911814886753078920583876993634, −7.10075715077162842875830829203, −6.79989588799441857040848321625, −5.82660136732763161536209615136, −5.27289291138736477083168630756, −4.10335085241219726420427450509, −3.60375309521087566139096355299, −2.62385752105451812353665252685, −2.13678824245385146321483853367, −0.62022020260467284746473901043, 0.62022020260467284746473901043, 2.13678824245385146321483853367, 2.62385752105451812353665252685, 3.60375309521087566139096355299, 4.10335085241219726420427450509, 5.27289291138736477083168630756, 5.82660136732763161536209615136, 6.79989588799441857040848321625, 7.10075715077162842875830829203, 7.911814886753078920583876993634

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