Properties

Label 2-60984-1.1-c1-0-21
Degree $2$
Conductor $60984$
Sign $1$
Analytic cond. $486.959$
Root an. cond. $22.0671$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s − 2·13-s + 4·17-s + 4·19-s + 6·23-s − 25-s + 2·29-s − 2·31-s − 2·35-s + 2·37-s + 6·43-s + 12·47-s + 49-s + 12·59-s + 2·61-s + 4·65-s + 4·67-s + 6·71-s − 10·73-s + 16·79-s + 6·83-s − 8·85-s + 6·89-s − 2·91-s − 8·95-s − 6·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 0.554·13-s + 0.970·17-s + 0.917·19-s + 1.25·23-s − 1/5·25-s + 0.371·29-s − 0.359·31-s − 0.338·35-s + 0.328·37-s + 0.914·43-s + 1.75·47-s + 1/7·49-s + 1.56·59-s + 0.256·61-s + 0.496·65-s + 0.488·67-s + 0.712·71-s − 1.17·73-s + 1.80·79-s + 0.658·83-s − 0.867·85-s + 0.635·89-s − 0.209·91-s − 0.820·95-s − 0.609·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60984\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(486.959\)
Root analytic conductor: \(22.0671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{60984} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 60984,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.507987110\)
\(L(\frac12)\) \(\approx\) \(2.507987110\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.36688614091338, −13.85417065446383, −13.30726445661809, −12.65068885701993, −12.12934742537017, −11.91286611739166, −11.22408956607150, −10.89042219682029, −10.22527677757610, −9.659192395809082, −9.173688380214732, −8.558503058594676, −7.992764901079372, −7.379135878058741, −7.330775646137443, −6.491140272730176, −5.655688064280380, −5.267382409784044, −4.685982838471223, −3.972057352673995, −3.515973873602858, −2.805675112796777, −2.175969381511351, −1.094153696627846, −0.6436710329112358, 0.6436710329112358, 1.094153696627846, 2.175969381511351, 2.805675112796777, 3.515973873602858, 3.972057352673995, 4.685982838471223, 5.267382409784044, 5.655688064280380, 6.491140272730176, 7.330775646137443, 7.379135878058741, 7.992764901079372, 8.558503058594676, 9.173688380214732, 9.659192395809082, 10.22527677757610, 10.89042219682029, 11.22408956607150, 11.91286611739166, 12.12934742537017, 12.65068885701993, 13.30726445661809, 13.85417065446383, 14.36688614091338

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