Properties

Label 2-60984-1.1-c1-0-3
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

\(L(1)\) \(\approx\) \(1.019371594\)
\(L(\frac12)\) \(\approx\) \(1.019371594\)
\(L(\frac{3}{2})\) 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

−14.37297072927481, −13.64801444158077, −13.25017932353968, −12.80373335771785, −12.33925689923017, −11.67675314227985, −11.19302465984713, −10.90976202809078, −10.31118695627771, −9.635716986889130, −9.035931308123467, −8.623677526038683, −8.167608079719018, −7.486844974138103, −6.917205399290138, −6.589527427642484, −5.840824394347271, −5.258179020804662, −4.527618059967795, −3.952568354272269, −3.629129515578905, −2.750087649170404, −2.193460246594804, −1.235401298611352, −0.3638282632530825, 0.3638282632530825, 1.235401298611352, 2.193460246594804, 2.750087649170404, 3.629129515578905, 3.952568354272269, 4.527618059967795, 5.258179020804662, 5.840824394347271, 6.589527427642484, 6.917205399290138, 7.486844974138103, 8.167608079719018, 8.623677526038683, 9.035931308123467, 9.635716986889130, 10.31118695627771, 10.90976202809078, 11.19302465984713, 11.67675314227985, 12.33925689923017, 12.80373335771785, 13.25017932353968, 13.64801444158077, 14.37297072927481

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