Properties

Label 2-60984-1.1-c1-0-30
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 − 5·13-s − 6·17-s + 8·19-s − 23-s − 25-s + 5·29-s + 5·31-s − 2·35-s − 8·37-s + 3·41-s + 43-s + 2·47-s + 49-s + 10·53-s − 3·59-s + 3·61-s + 10·65-s + 13·67-s − 15·71-s − 12·73-s − 8·79-s − 11·83-s + 12·85-s − 15·89-s − 5·91-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 1.38·13-s − 1.45·17-s + 1.83·19-s − 0.208·23-s − 1/5·25-s + 0.928·29-s + 0.898·31-s − 0.338·35-s − 1.31·37-s + 0.468·41-s + 0.152·43-s + 0.291·47-s + 1/7·49-s + 1.37·53-s − 0.390·59-s + 0.384·61-s + 1.24·65-s + 1.58·67-s − 1.78·71-s − 1.40·73-s − 0.900·79-s − 1.20·83-s + 1.30·85-s − 1.58·89-s − 0.524·91-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 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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

−14.38875252971189, −14.14101045258278, −13.64691159227608, −12.99078611854161, −12.40436352970339, −11.88013311372616, −11.59789483000676, −11.24910211454873, −10.26728308799274, −10.13140870673626, −9.435405561477857, −8.709355534675475, −8.463541476135215, −7.646959388802785, −7.273313624141618, −6.984423949731550, −6.111515728938848, −5.434874923207817, −4.853465844626477, −4.409660109685256, −3.859109661263252, −2.990737052638652, −2.563172535082158, −1.731990284992748, −0.8012203782857740, 0, 0.8012203782857740, 1.731990284992748, 2.563172535082158, 2.990737052638652, 3.859109661263252, 4.409660109685256, 4.853465844626477, 5.434874923207817, 6.111515728938848, 6.984423949731550, 7.273313624141618, 7.646959388802785, 8.463541476135215, 8.709355534675475, 9.435405561477857, 10.13140870673626, 10.26728308799274, 11.24910211454873, 11.59789483000676, 11.88013311372616, 12.40436352970339, 12.99078611854161, 13.64691159227608, 14.14101045258278, 14.38875252971189

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