Properties

Label 2-33600-1.1-c1-0-98
Degree $2$
Conductor $33600$
Sign $-1$
Analytic cond. $268.297$
Root an. cond. $16.3797$
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 + 7-s + 9-s − 4·11-s − 6·13-s − 6·17-s − 21-s + 4·23-s − 27-s + 6·29-s + 4·33-s + 2·37-s + 6·39-s + 2·41-s + 4·43-s − 4·47-s + 49-s + 6·51-s − 6·53-s − 12·59-s + 10·61-s + 63-s + 4·67-s − 4·69-s + 8·71-s + 14·73-s − 4·77-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 1.45·17-s − 0.218·21-s + 0.834·23-s − 0.192·27-s + 1.11·29-s + 0.696·33-s + 0.328·37-s + 0.960·39-s + 0.312·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 1.56·59-s + 1.28·61-s + 0.125·63-s + 0.488·67-s − 0.481·69-s + 0.949·71-s + 1.63·73-s − 0.455·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(268.297\)
Root analytic conductor: \(16.3797\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{33600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 33600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−15.44573536626804, −14.79819158624228, −14.20464821117267, −13.69202598543361, −12.94084787632240, −12.70610349798277, −12.14216451609336, −11.50501744123803, −10.89152082457749, −10.71077106905316, −9.893135802671519, −9.518201913631928, −8.825463700301047, −8.031405889666433, −7.754941866074965, −6.865150062011028, −6.709315946211423, −5.745677397252429, −5.125299018015288, −4.734857782499968, −4.307036724312266, −3.146163093724765, −2.489841303996047, −2.010590347150409, −0.8003610766700175, 0, 0.8003610766700175, 2.010590347150409, 2.489841303996047, 3.146163093724765, 4.307036724312266, 4.734857782499968, 5.125299018015288, 5.745677397252429, 6.709315946211423, 6.865150062011028, 7.754941866074965, 8.031405889666433, 8.825463700301047, 9.518201913631928, 9.893135802671519, 10.71077106905316, 10.89152082457749, 11.50501744123803, 12.14216451609336, 12.70610349798277, 12.94084787632240, 13.69202598543361, 14.20464821117267, 14.79819158624228, 15.44573536626804

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