Properties

Label 2-864-1.1-c1-0-3
Degree $2$
Conductor $864$
Sign $1$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 3·7-s − 3·11-s + 4·17-s + 6·19-s + 6·23-s − 4·25-s − 2·29-s + 9·31-s − 3·35-s − 2·37-s − 10·41-s + 6·43-s + 6·47-s + 2·49-s + 13·53-s + 3·55-s − 12·59-s + 8·61-s + 6·67-s + 12·71-s + 9·73-s − 9·77-s − 3·83-s − 4·85-s + 14·89-s − 6·95-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.13·7-s − 0.904·11-s + 0.970·17-s + 1.37·19-s + 1.25·23-s − 4/5·25-s − 0.371·29-s + 1.61·31-s − 0.507·35-s − 0.328·37-s − 1.56·41-s + 0.914·43-s + 0.875·47-s + 2/7·49-s + 1.78·53-s + 0.404·55-s − 1.56·59-s + 1.02·61-s + 0.733·67-s + 1.42·71-s + 1.05·73-s − 1.02·77-s − 0.329·83-s − 0.433·85-s + 1.48·89-s − 0.615·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.643655999\)
\(L(\frac12)\) \(\approx\) \(1.643655999\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−10.24151483342147504043812250434, −9.344827270962992036294829473440, −8.180464945333468393072503118033, −7.82476086711398995912338456757, −6.93499396476405803707004808409, −5.45678967002093979727974577207, −5.03134570361553185426866134499, −3.77979388504550471995644973896, −2.63884396406011957309091091690, −1.11568290382473683806344984693, 1.11568290382473683806344984693, 2.63884396406011957309091091690, 3.77979388504550471995644973896, 5.03134570361553185426866134499, 5.45678967002093979727974577207, 6.93499396476405803707004808409, 7.82476086711398995912338456757, 8.180464945333468393072503118033, 9.344827270962992036294829473440, 10.24151483342147504043812250434

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