Properties

Label 2-4032-1.1-c1-0-48
Degree $2$
Conductor $4032$
Sign $-1$
Analytic cond. $32.1956$
Root an. cond. $5.67412$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 4·13-s − 6·17-s − 2·19-s − 5·25-s − 6·29-s − 4·31-s − 2·37-s − 6·41-s − 8·43-s + 12·47-s + 49-s + 6·53-s − 6·59-s − 8·61-s + 4·67-s + 2·73-s + 8·79-s − 6·83-s + 6·89-s + 4·91-s − 10·97-s − 4·103-s + 12·107-s − 2·109-s − 6·113-s − 6·119-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.10·13-s − 1.45·17-s − 0.458·19-s − 25-s − 1.11·29-s − 0.718·31-s − 0.328·37-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 0.781·59-s − 1.02·61-s + 0.488·67-s + 0.234·73-s + 0.900·79-s − 0.658·83-s + 0.635·89-s + 0.419·91-s − 1.01·97-s − 0.394·103-s + 1.16·107-s − 0.191·109-s − 0.564·113-s − 0.550·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4032} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ -1)\)

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

−8.156759517304686892380266927456, −7.34808391035356365236166309157, −6.58338612370338623267809377565, −5.89372115656638311788103577429, −5.10276803747253805923263658121, −4.14318817559479393176326791421, −3.59991440544977347073267949607, −2.31147598100149017773134041140, −1.55194931839012879683964780078, 0, 1.55194931839012879683964780078, 2.31147598100149017773134041140, 3.59991440544977347073267949607, 4.14318817559479393176326791421, 5.10276803747253805923263658121, 5.89372115656638311788103577429, 6.58338612370338623267809377565, 7.34808391035356365236166309157, 8.156759517304686892380266927456

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