↑

Properties

Label 2-32192-1.1-c1-0-16
Degree $2$
Conductor $32192$
Sign $-1$
Analytic cond. $257.054$
Root an. cond. $16.0329$
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 + 3·7-s − 2·9-s + 3·11-s + 13-s − 6·17-s − 4·19-s − 3·21-s + 7·23-s − 5·25-s + 5·27-s − 4·29-s + 4·31-s − 3·33-s + 2·37-s − 39-s − 8·41-s − 43-s + 9·47-s + 2·49-s + 6·51-s − 2·53-s + 4·57-s − 4·59-s + 5·61-s − 6·63-s + 7·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s − 2/3·9-s + 0.904·11-s + 0.277·13-s − 1.45·17-s − 0.917·19-s − 0.654·21-s + 1.45·23-s − 25-s + 0.962·27-s − 0.742·29-s + 0.718·31-s − 0.522·33-s + 0.328·37-s − 0.160·39-s − 1.24·41-s − 0.152·43-s + 1.31·47-s + 2/7·49-s + 0.840·51-s − 0.274·53-s + 0.529·57-s − 0.520·59-s + 0.640·61-s − 0.755·63-s + 0.855·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32192\)    =    \(2^{6} \cdot 503\)
Sign: $-1$
Analytic conductor: \(257.054\)
Root analytic conductor: \(16.0329\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 32192,\ (\ :1/2),\ -1)\)

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 \( 1 \)
503 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−15.30933951273478, −14.83082065107446, −14.25328395578763, −13.82384391044050, −13.17353587355449, −12.71761788393899, −11.78481383743145, −11.64097972446783, −11.11192147191580, −10.78165711162885, −10.09232628072472, −9.185893227902384, −8.802051340352409, −8.419884187392046, −7.707320181203985, −6.949932462572735, −6.474076557209680, −5.933580251700791, −5.241837266430944, −4.653137040543542, −4.196127448105049, −3.396822015034972, −2.444105164441875, −1.830202048912137, −1.005505170504542, 0, 1.005505170504542, 1.830202048912137, 2.444105164441875, 3.396822015034972, 4.196127448105049, 4.653137040543542, 5.241837266430944, 5.933580251700791, 6.474076557209680, 6.949932462572735, 7.707320181203985, 8.419884187392046, 8.802051340352409, 9.185893227902384, 10.09232628072472, 10.78165711162885, 11.11192147191580, 11.64097972446783, 11.78481383743145, 12.71761788393899, 13.17353587355449, 13.82384391044050, 14.25328395578763, 14.83082065107446, 15.30933951273478

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