Properties

Label 2-6864-1.1-c1-0-16
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 − 1.05·5-s − 4.24·7-s + 9-s + 11-s + 13-s − 1.05·15-s − 4.55·17-s + 0.977·19-s − 4.24·21-s + 2.82·23-s − 3.88·25-s + 27-s + 4.75·29-s + 8.28·31-s + 33-s + 4.47·35-s − 6.14·37-s + 39-s − 6.92·41-s − 7.56·43-s − 1.05·45-s − 0.967·47-s + 11.0·49-s − 4.55·51-s − 5.76·53-s − 1.05·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.471·5-s − 1.60·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s − 0.272·15-s − 1.10·17-s + 0.224·19-s − 0.927·21-s + 0.589·23-s − 0.777·25-s + 0.192·27-s + 0.882·29-s + 1.48·31-s + 0.174·33-s + 0.756·35-s − 1.00·37-s + 0.160·39-s − 1.08·41-s − 1.15·43-s − 0.157·45-s − 0.141·47-s + 1.57·49-s − 0.637·51-s − 0.791·53-s − 0.142·55-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

\(L(1)\) \(\approx\) \(1.454810130\)
\(L(\frac12)\) \(\approx\) \(1.454810130\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.05T + 5T^{2} \)
7 \( 1 + 4.24T + 7T^{2} \)
17 \( 1 + 4.55T + 17T^{2} \)
19 \( 1 - 0.977T + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 - 4.75T + 29T^{2} \)
31 \( 1 - 8.28T + 31T^{2} \)
37 \( 1 + 6.14T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 + 7.56T + 43T^{2} \)
47 \( 1 + 0.967T + 47T^{2} \)
53 \( 1 + 5.76T + 53T^{2} \)
59 \( 1 - 5.16T + 59T^{2} \)
61 \( 1 - 12.8T + 61T^{2} \)
67 \( 1 + 8.76T + 67T^{2} \)
71 \( 1 + 3.74T + 71T^{2} \)
73 \( 1 - 0.141T + 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 - 8.73T + 83T^{2} \)
89 \( 1 + 16.5T + 89T^{2} \)
97 \( 1 + 7.42T + 97T^{2} \)
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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

−8.115245408550987433438514186799, −7.06682787137527901240691635534, −6.69177197386066263626764891323, −6.12074701342741722990695772413, −4.99564301928884440720064819931, −4.20262397782384183817420275541, −3.41902660149345206185323283447, −2.98523536701454257071044742742, −1.93966225224493082027308560893, −0.57846996500935879179241950659, 0.57846996500935879179241950659, 1.93966225224493082027308560893, 2.98523536701454257071044742742, 3.41902660149345206185323283447, 4.20262397782384183817420275541, 4.99564301928884440720064819931, 6.12074701342741722990695772413, 6.69177197386066263626764891323, 7.06682787137527901240691635534, 8.115245408550987433438514186799

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