Properties

Label 2-864-1.1-c1-0-14
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 $1$

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: \(1\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−9.622687493624043148972478109216, −9.103676153893653346174230076714, −8.096235085912272934101439989171, −7.00681796643489856555968651557, −6.31855619934285825414386333790, −5.42474736940027213899015616950, −4.30284121931526434248881423255, −3.10596268507022717082044228879, −2.09771190244944839202554321983, 0, 2.09771190244944839202554321983, 3.10596268507022717082044228879, 4.30284121931526434248881423255, 5.42474736940027213899015616950, 6.31855619934285825414386333790, 7.00681796643489856555968651557, 8.096235085912272934101439989171, 9.103676153893653346174230076714, 9.622687493624043148972478109216

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