Properties

Label 2-216384-1.1-c1-0-195
Degree $2$
Conductor $216384$
Sign $-1$
Analytic cond. $1727.83$
Root an. cond. $41.5672$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·5-s + 9-s + 4·11-s − 3·13-s − 3·15-s + 4·17-s − 23-s + 4·25-s − 27-s − 3·29-s − 6·31-s − 4·33-s + 9·37-s + 3·39-s − 9·41-s − 3·43-s + 3·45-s − 7·47-s − 4·51-s + 4·53-s + 12·55-s − 6·59-s + 10·61-s − 9·65-s + 4·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 1/3·9-s + 1.20·11-s − 0.832·13-s − 0.774·15-s + 0.970·17-s − 0.208·23-s + 4/5·25-s − 0.192·27-s − 0.557·29-s − 1.07·31-s − 0.696·33-s + 1.47·37-s + 0.480·39-s − 1.40·41-s − 0.457·43-s + 0.447·45-s − 1.02·47-s − 0.560·51-s + 0.549·53-s + 1.61·55-s − 0.781·59-s + 1.28·61-s − 1.11·65-s + 0.488·67-s + 0.120·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216384\)    =    \(2^{6} \cdot 3 \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(1727.83\)
Root analytic conductor: \(41.5672\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{216384} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 216384,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
23 \( 1 + T \)
good5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.12091424060902, −12.77446037326404, −12.33562313148825, −11.72991725240107, −11.46836191778377, −10.93739290881852, −10.20827649408135, −9.908309658312781, −9.651721288533216, −9.143133198810832, −8.655627947206124, −7.915301673615536, −7.454919940694514, −6.861148858693758, −6.413222862949279, −6.068650304411520, −5.413822333426500, −5.180828099318172, −4.578819955875465, −3.789693883394012, −3.453857067177777, −2.563127544474797, −2.013934427662691, −1.502612890333851, −0.9414431450082615, 0, 0.9414431450082615, 1.502612890333851, 2.013934427662691, 2.563127544474797, 3.453857067177777, 3.789693883394012, 4.578819955875465, 5.180828099318172, 5.413822333426500, 6.068650304411520, 6.413222862949279, 6.861148858693758, 7.454919940694514, 7.915301673615536, 8.655627947206124, 9.143133198810832, 9.651721288533216, 9.908309658312781, 10.20827649408135, 10.93739290881852, 11.46836191778377, 11.72991725240107, 12.33562313148825, 12.77446037326404, 13.12091424060902

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