Properties

Label 2-6864-1.1-c1-0-51
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 + 0.279·5-s + 4.36·7-s + 9-s − 11-s − 13-s + 0.279·15-s + 2.64·17-s − 4.58·19-s + 4.36·21-s − 1.80·23-s − 4.92·25-s + 27-s + 3.30·29-s + 1.50·31-s − 33-s + 1.22·35-s − 0.559·37-s − 39-s + 6.36·41-s + 6.97·43-s + 0.279·45-s − 2.73·47-s + 12.0·49-s + 2.64·51-s + 2.99·53-s − 0.279·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.125·5-s + 1.65·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s + 0.0722·15-s + 0.641·17-s − 1.05·19-s + 0.952·21-s − 0.376·23-s − 0.984·25-s + 0.192·27-s + 0.614·29-s + 0.269·31-s − 0.174·33-s + 0.206·35-s − 0.0920·37-s − 0.160·39-s + 0.994·41-s + 1.06·43-s + 0.0417·45-s − 0.398·47-s + 1.72·49-s + 0.370·51-s + 0.411·53-s − 0.0377·55-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

\(L(1)\) \(\approx\) \(3.236680786\)
\(L(\frac12)\) \(\approx\) \(3.236680786\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.279T + 5T^{2} \)
7 \( 1 - 4.36T + 7T^{2} \)
17 \( 1 - 2.64T + 17T^{2} \)
19 \( 1 + 4.58T + 19T^{2} \)
23 \( 1 + 1.80T + 23T^{2} \)
29 \( 1 - 3.30T + 29T^{2} \)
31 \( 1 - 1.50T + 31T^{2} \)
37 \( 1 + 0.559T + 37T^{2} \)
41 \( 1 - 6.36T + 41T^{2} \)
43 \( 1 - 6.97T + 43T^{2} \)
47 \( 1 + 2.73T + 47T^{2} \)
53 \( 1 - 2.99T + 53T^{2} \)
59 \( 1 - 8.39T + 59T^{2} \)
61 \( 1 + 1.10T + 61T^{2} \)
67 \( 1 - 4.39T + 67T^{2} \)
71 \( 1 - 2.99T + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 2.49T + 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−8.062503805551354975770420010132, −7.54500046320063816238303909798, −6.64018581090668569659762554102, −5.74497649761612070621231836771, −5.05565054609485829651572473322, −4.36176015423648488926371019166, −3.70104113418123551034458910891, −2.44074017873941600848111925102, −2.01779302182352517373450885736, −0.923436639841302740364866937713, 0.923436639841302740364866937713, 2.01779302182352517373450885736, 2.44074017873941600848111925102, 3.70104113418123551034458910891, 4.36176015423648488926371019166, 5.05565054609485829651572473322, 5.74497649761612070621231836771, 6.64018581090668569659762554102, 7.54500046320063816238303909798, 8.062503805551354975770420010132

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