Properties

Label 2-6864-1.1-c1-0-68
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 + 3.52·5-s + 3.24·7-s + 9-s + 11-s + 13-s + 3.52·15-s − 5.48·17-s + 2.44·19-s + 3.24·21-s − 2.29·23-s + 7.45·25-s + 27-s + 9.32·29-s − 2.32·31-s + 33-s + 11.4·35-s − 11.4·37-s + 39-s − 10.0·41-s + 7.09·43-s + 3.52·45-s + 2.48·47-s + 3.55·49-s − 5.48·51-s + 13.6·53-s + 3.52·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.57·5-s + 1.22·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 0.911·15-s − 1.33·17-s + 0.561·19-s + 0.708·21-s − 0.478·23-s + 1.49·25-s + 0.192·27-s + 1.73·29-s − 0.418·31-s + 0.174·33-s + 1.93·35-s − 1.88·37-s + 0.160·39-s − 1.57·41-s + 1.08·43-s + 0.526·45-s + 0.362·47-s + 0.507·49-s − 0.768·51-s + 1.87·53-s + 0.475·55-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

\(L(1)\) \(\approx\) \(4.394492410\)
\(L(\frac12)\) \(\approx\) \(4.394492410\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.52T + 5T^{2} \)
7 \( 1 - 3.24T + 7T^{2} \)
17 \( 1 + 5.48T + 17T^{2} \)
19 \( 1 - 2.44T + 19T^{2} \)
23 \( 1 + 2.29T + 23T^{2} \)
29 \( 1 - 9.32T + 29T^{2} \)
31 \( 1 + 2.32T + 31T^{2} \)
37 \( 1 + 11.4T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 7.09T + 43T^{2} \)
47 \( 1 - 2.48T + 47T^{2} \)
53 \( 1 - 13.6T + 53T^{2} \)
59 \( 1 - 5.35T + 59T^{2} \)
61 \( 1 + 7.71T + 61T^{2} \)
67 \( 1 - 12.2T + 67T^{2} \)
71 \( 1 + 1.76T + 71T^{2} \)
73 \( 1 - 1.80T + 73T^{2} \)
79 \( 1 + 3.26T + 79T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 + 2.06T + 89T^{2} \)
97 \( 1 + 5.04T + 97T^{2} \)
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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

−8.225325549764904033650524953708, −7.11949879599935645799159087250, −6.66458309020129554889995246029, −5.79279225924120180045012719339, −5.12018757089083878180399294952, −4.51128744757934121803780445878, −3.52760204514209158790318238752, −2.39906283848335960922679132612, −1.95106675103046680636330258061, −1.14364279698951483970563004717, 1.14364279698951483970563004717, 1.95106675103046680636330258061, 2.39906283848335960922679132612, 3.52760204514209158790318238752, 4.51128744757934121803780445878, 5.12018757089083878180399294952, 5.79279225924120180045012719339, 6.66458309020129554889995246029, 7.11949879599935645799159087250, 8.225325549764904033650524953708

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