Properties

Label 2-60984-1.1-c1-0-52
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 7-s − 2·13-s − 6·17-s + 4·19-s + 6·23-s − 25-s − 4·31-s + 2·35-s + 10·37-s − 2·41-s + 4·43-s + 4·47-s + 49-s − 12·53-s + 12·59-s − 6·61-s − 4·65-s − 4·67-s − 14·71-s + 2·73-s + 8·79-s + 16·83-s − 12·85-s − 6·89-s − 2·91-s + 8·95-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 1.25·23-s − 1/5·25-s − 0.718·31-s + 0.338·35-s + 1.64·37-s − 0.312·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s − 1.64·53-s + 1.56·59-s − 0.768·61-s − 0.496·65-s − 0.488·67-s − 1.66·71-s + 0.234·73-s + 0.900·79-s + 1.75·83-s − 1.30·85-s − 0.635·89-s − 0.209·91-s + 0.820·95-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: \(1\)
Selberg data: \((2,\ 60984,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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

−14.65412440255806, −13.88520318416609, −13.57522718012505, −13.14534156611942, −12.60829791782300, −12.05680550298898, −11.30690444045029, −11.10099975784783, −10.52717120786464, −9.820618888110479, −9.392726136006784, −9.062061918421847, −8.450734123127559, −7.595124486133009, −7.411475018049278, −6.534561584030026, −6.251305134545225, −5.376875980662044, −5.129444045006680, −4.411828350121286, −3.841922805126475, −2.808935235171390, −2.522119000659973, −1.713480180980772, −1.079822417489575, 0, 1.079822417489575, 1.713480180980772, 2.522119000659973, 2.808935235171390, 3.841922805126475, 4.411828350121286, 5.129444045006680, 5.376875980662044, 6.251305134545225, 6.534561584030026, 7.411475018049278, 7.595124486133009, 8.450734123127559, 9.062061918421847, 9.392726136006784, 9.820618888110479, 10.52717120786464, 11.10099975784783, 11.30690444045029, 12.05680550298898, 12.60829791782300, 13.14534156611942, 13.57522718012505, 13.88520318416609, 14.65412440255806

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