Properties

Label 2-1440-1.1-c3-0-7
Degree $2$
Conductor $1440$
Sign $1$
Analytic cond. $84.9627$
Root an. cond. $9.21752$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s − 30·7-s + 50·11-s − 88·13-s − 74·17-s − 140·19-s + 80·23-s + 25·25-s + 234·29-s − 150·35-s + 116·37-s + 72·41-s − 280·43-s + 120·47-s + 557·49-s + 498·53-s + 250·55-s − 870·59-s + 650·61-s − 440·65-s − 420·67-s + 1.02e3·71-s − 322·73-s − 1.50e3·77-s − 160·79-s − 980·83-s − 370·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.61·7-s + 1.37·11-s − 1.87·13-s − 1.05·17-s − 1.69·19-s + 0.725·23-s + 1/5·25-s + 1.49·29-s − 0.724·35-s + 0.515·37-s + 0.274·41-s − 0.993·43-s + 0.372·47-s + 1.62·49-s + 1.29·53-s + 0.612·55-s − 1.91·59-s + 1.36·61-s − 0.839·65-s − 0.765·67-s + 1.70·71-s − 0.516·73-s − 2.22·77-s − 0.227·79-s − 1.29·83-s − 0.472·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(84.9627\)
Root analytic conductor: \(9.21752\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.269597898\)
\(L(\frac12)\) \(\approx\) \(1.269597898\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5 \( 1 - p T \)
good7 \( 1 + 30 T + p^{3} T^{2} \)
11 \( 1 - 50 T + p^{3} T^{2} \)
13 \( 1 + 88 T + p^{3} T^{2} \)
17 \( 1 + 74 T + p^{3} T^{2} \)
19 \( 1 + 140 T + p^{3} T^{2} \)
23 \( 1 - 80 T + p^{3} T^{2} \)
29 \( 1 - 234 T + p^{3} T^{2} \)
31 \( 1 + p^{3} T^{2} \)
37 \( 1 - 116 T + p^{3} T^{2} \)
41 \( 1 - 72 T + p^{3} T^{2} \)
43 \( 1 + 280 T + p^{3} T^{2} \)
47 \( 1 - 120 T + p^{3} T^{2} \)
53 \( 1 - 498 T + p^{3} T^{2} \)
59 \( 1 + 870 T + p^{3} T^{2} \)
61 \( 1 - 650 T + p^{3} T^{2} \)
67 \( 1 + 420 T + p^{3} T^{2} \)
71 \( 1 - 1020 T + p^{3} T^{2} \)
73 \( 1 + 322 T + p^{3} T^{2} \)
79 \( 1 + 160 T + p^{3} T^{2} \)
83 \( 1 + 980 T + p^{3} T^{2} \)
89 \( 1 - 1124 T + p^{3} T^{2} \)
97 \( 1 - 1114 T + p^{3} T^{2} \)
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

−9.170770449175621346641164037392, −8.698067835935893108881640921367, −7.25145577137105647403766833281, −6.56085985836188356127498568580, −6.25861228117705552377472002338, −4.86388977716025109580838227857, −4.09493565939268740159186022999, −2.91591103789219758503069008077, −2.14227206263947767241181183518, −0.52728645845441789911851267592, 0.52728645845441789911851267592, 2.14227206263947767241181183518, 2.91591103789219758503069008077, 4.09493565939268740159186022999, 4.86388977716025109580838227857, 6.25861228117705552377472002338, 6.56085985836188356127498568580, 7.25145577137105647403766833281, 8.698067835935893108881640921367, 9.170770449175621346641164037392

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