Properties

Degree $2$
Conductor $86640$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 2·7-s + 9-s + 2·13-s − 15-s − 2·17-s + 2·21-s + 2·23-s + 25-s + 27-s − 4·29-s + 4·31-s − 2·35-s + 2·37-s + 2·39-s + 4·41-s − 10·43-s − 45-s − 6·47-s − 3·49-s − 2·51-s + 6·53-s − 4·59-s − 10·61-s + 2·63-s − 2·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.436·21-s + 0.417·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.718·31-s − 0.338·35-s + 0.328·37-s + 0.320·39-s + 0.624·41-s − 1.52·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s − 0.520·59-s − 1.28·61-s + 0.251·63-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(86640\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{86640} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 86640,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
19 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.10494860585205, −13.71738350319143, −13.17917640152698, −12.81718743742997, −12.15640344827018, −11.50627968958853, −11.35064756676668, −10.70405453666816, −10.22352189383803, −9.587688263673400, −9.028683874539496, −8.580214703913577, −8.161168683997981, −7.614107207429696, −7.233096609712540, −6.439657064506731, −6.099820117997387, −5.158589016211476, −4.767552473532970, −4.220120196203279, −3.559291685789429, −3.081023805656040, −2.312349794880842, −1.662535830899125, −1.040799928106296, 0, 1.040799928106296, 1.662535830899125, 2.312349794880842, 3.081023805656040, 3.559291685789429, 4.220120196203279, 4.767552473532970, 5.158589016211476, 6.099820117997387, 6.439657064506731, 7.233096609712540, 7.614107207429696, 8.161168683997981, 8.580214703913577, 9.028683874539496, 9.587688263673400, 10.22352189383803, 10.70405453666816, 11.35064756676668, 11.50627968958853, 12.15640344827018, 12.81718743742997, 13.17917640152698, 13.71738350319143, 14.10494860585205

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