Properties

Label 2-270-1.1-c3-0-8
Degree $2$
Conductor $270$
Sign $1$
Analytic cond. $15.9305$
Root an. cond. $3.99130$
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  + 2·2-s + 4·4-s + 5·5-s + 8·7-s + 8·8-s + 10·10-s + 18·11-s + 8·13-s + 16·14-s + 16·16-s + 15·17-s + 23·19-s + 20·20-s + 36·22-s + 63·23-s + 25·25-s + 16·26-s + 32·28-s + 156·29-s − 85·31-s + 32·32-s + 30·34-s + 40·35-s + 74·37-s + 46·38-s + 40·40-s + 246·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.431·7-s + 0.353·8-s + 0.316·10-s + 0.493·11-s + 0.170·13-s + 0.305·14-s + 1/4·16-s + 0.214·17-s + 0.277·19-s + 0.223·20-s + 0.348·22-s + 0.571·23-s + 1/5·25-s + 0.120·26-s + 0.215·28-s + 0.998·29-s − 0.492·31-s + 0.176·32-s + 0.151·34-s + 0.193·35-s + 0.328·37-s + 0.196·38-s + 0.158·40-s + 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(15.9305\)
Root analytic conductor: \(3.99130\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.342071040\)
\(L(\frac12)\) \(\approx\) \(3.342071040\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 \)
5 \( 1 - p T \)
good7 \( 1 - 8 T + p^{3} T^{2} \)
11 \( 1 - 18 T + p^{3} T^{2} \)
13 \( 1 - 8 T + p^{3} T^{2} \)
17 \( 1 - 15 T + p^{3} T^{2} \)
19 \( 1 - 23 T + p^{3} T^{2} \)
23 \( 1 - 63 T + p^{3} T^{2} \)
29 \( 1 - 156 T + p^{3} T^{2} \)
31 \( 1 + 85 T + p^{3} T^{2} \)
37 \( 1 - 2 p T + p^{3} T^{2} \)
41 \( 1 - 6 p T + p^{3} T^{2} \)
43 \( 1 + 190 T + p^{3} T^{2} \)
47 \( 1 - 288 T + p^{3} T^{2} \)
53 \( 1 + 177 T + p^{3} T^{2} \)
59 \( 1 - 792 T + p^{3} T^{2} \)
61 \( 1 + 907 T + p^{3} T^{2} \)
67 \( 1 + 322 T + p^{3} T^{2} \)
71 \( 1 + 270 T + p^{3} T^{2} \)
73 \( 1 - 254 T + p^{3} T^{2} \)
79 \( 1 + 1123 T + p^{3} T^{2} \)
83 \( 1 + 771 T + p^{3} T^{2} \)
89 \( 1 + 198 T + p^{3} T^{2} \)
97 \( 1 + 1192 T + p^{3} T^{2} \)
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

−11.57001951013826554793011426862, −10.72707780336065976470252948172, −9.661628232205811914802586376075, −8.572816793186980572462505611973, −7.37809886026688680309763393915, −6.33358969985050718352326282398, −5.32818794579185470192040830564, −4.24491218427357514486926764281, −2.87345634784495729641931328210, −1.38041783395827634040710503882, 1.38041783395827634040710503882, 2.87345634784495729641931328210, 4.24491218427357514486926764281, 5.32818794579185470192040830564, 6.33358969985050718352326282398, 7.37809886026688680309763393915, 8.572816793186980572462505611973, 9.661628232205811914802586376075, 10.72707780336065976470252948172, 11.57001951013826554793011426862

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