Properties

Label 2-102960-1.1-c1-0-122
Degree $2$
Conductor $102960$
Sign $1$
Analytic cond. $822.139$
Root an. cond. $28.6729$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s − 11-s + 13-s − 2·17-s − 4·19-s + 25-s − 2·29-s − 4·31-s + 4·35-s − 6·37-s − 2·41-s − 4·43-s + 9·49-s + 6·53-s + 55-s − 4·59-s − 2·61-s − 65-s + 12·67-s + 10·73-s + 4·77-s + 8·79-s − 12·83-s + 2·85-s + 6·89-s − 4·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s − 0.301·11-s + 0.277·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.676·35-s − 0.986·37-s − 0.312·41-s − 0.609·43-s + 9/7·49-s + 0.824·53-s + 0.134·55-s − 0.520·59-s − 0.256·61-s − 0.124·65-s + 1.46·67-s + 1.17·73-s + 0.455·77-s + 0.900·79-s − 1.31·83-s + 0.216·85-s + 0.635·89-s − 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102960\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(822.139\)
Root analytic conductor: \(28.6729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 102960,\ (\ :1/2),\ 1)\)

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 \( 1 \)
5 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−14.09693545163895, −13.59653583018554, −13.20912761701669, −12.73785666813590, −12.36669623219525, −11.90695771164521, −11.15052154135898, −10.79432412920144, −10.34532692467666, −9.709735685092446, −9.346346254024201, −8.747454793748052, −8.333300447513373, −7.728959923528519, −7.027388517091630, −6.661872038442897, −6.317287671699815, −5.512777481533317, −5.142271065129858, −4.236875031310579, −3.795849618303774, −3.361337446071899, −2.651510714765202, −2.099477065200590, −1.177988436113182, 0, 0, 1.177988436113182, 2.099477065200590, 2.651510714765202, 3.361337446071899, 3.795849618303774, 4.236875031310579, 5.142271065129858, 5.512777481533317, 6.317287671699815, 6.661872038442897, 7.027388517091630, 7.728959923528519, 8.333300447513373, 8.747454793748052, 9.346346254024201, 9.709735685092446, 10.34532692467666, 10.79432412920144, 11.15052154135898, 11.90695771164521, 12.36669623219525, 12.73785666813590, 13.20912761701669, 13.59653583018554, 14.09693545163895

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