Properties

Label 2-6864-1.1-c1-0-59
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 + 1.82·5-s + 0.862·7-s + 9-s + 11-s + 13-s + 1.82·15-s + 0.551·17-s + 0.300·19-s + 0.862·21-s + 7.71·23-s − 1.65·25-s + 27-s − 4.56·29-s − 0.391·31-s + 33-s + 1.57·35-s + 9.60·37-s + 39-s + 5.29·41-s + 11.7·43-s + 1.82·45-s − 6.50·47-s − 6.25·49-s + 0.551·51-s − 9.76·53-s + 1.82·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.817·5-s + 0.326·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 0.472·15-s + 0.133·17-s + 0.0690·19-s + 0.188·21-s + 1.60·23-s − 0.330·25-s + 0.192·27-s − 0.847·29-s − 0.0702·31-s + 0.174·33-s + 0.266·35-s + 1.57·37-s + 0.160·39-s + 0.827·41-s + 1.79·43-s + 0.272·45-s − 0.949·47-s − 0.893·49-s + 0.0772·51-s − 1.34·53-s + 0.246·55-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

\(L(1)\) \(\approx\) \(3.532482064\)
\(L(\frac12)\) \(\approx\) \(3.532482064\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.82T + 5T^{2} \)
7 \( 1 - 0.862T + 7T^{2} \)
17 \( 1 - 0.551T + 17T^{2} \)
19 \( 1 - 0.300T + 19T^{2} \)
23 \( 1 - 7.71T + 23T^{2} \)
29 \( 1 + 4.56T + 29T^{2} \)
31 \( 1 + 0.391T + 31T^{2} \)
37 \( 1 - 9.60T + 37T^{2} \)
41 \( 1 - 5.29T + 41T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 + 6.50T + 47T^{2} \)
53 \( 1 + 9.76T + 53T^{2} \)
59 \( 1 - 6.49T + 59T^{2} \)
61 \( 1 + 7.39T + 61T^{2} \)
67 \( 1 + 8.53T + 67T^{2} \)
71 \( 1 - 3.58T + 71T^{2} \)
73 \( 1 - 0.795T + 73T^{2} \)
79 \( 1 - 9.81T + 79T^{2} \)
83 \( 1 + 5.49T + 83T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 - 2.57T + 97T^{2} \)
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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

−7.82391348596419459478227971082, −7.48421191885414531866865743446, −6.45232263660216577055565817176, −5.96753137440084031249613958387, −5.09252947622980778679508992214, −4.39099725110607790853493717177, −3.47904546285437507671587120762, −2.68631713333425508368937827088, −1.85117304742605595957822933251, −0.994888079861440283058567961533, 0.994888079861440283058567961533, 1.85117304742605595957822933251, 2.68631713333425508368937827088, 3.47904546285437507671587120762, 4.39099725110607790853493717177, 5.09252947622980778679508992214, 5.96753137440084031249613958387, 6.45232263660216577055565817176, 7.48421191885414531866865743446, 7.82391348596419459478227971082

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