↑

Properties

Label 2-129360-1.1-c1-0-11
Degree $2$
Conductor $129360$
Sign $1$
Analytic cond. $1032.94$
Root an. cond. $32.1394$
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  + 3-s − 5-s + 9-s + 11-s − 2·13-s − 15-s − 2·17-s − 8·23-s + 25-s + 27-s + 2·29-s − 8·31-s + 33-s + 6·37-s − 2·39-s + 2·41-s − 12·43-s − 45-s + 8·47-s − 2·51-s − 10·53-s − 55-s − 12·59-s + 2·61-s + 2·65-s − 4·67-s − 8·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.174·33-s + 0.986·37-s − 0.320·39-s + 0.312·41-s − 1.82·43-s − 0.149·45-s + 1.16·47-s − 0.280·51-s − 1.37·53-s − 0.134·55-s − 1.56·59-s + 0.256·61-s + 0.248·65-s − 0.488·67-s − 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129360\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(1032.94\)
Root analytic conductor: \(32.1394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 129360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.289216194\)
\(L(\frac12)\) \(\approx\) \(1.289216194\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.58781678031827, −12.95863179582281, −12.50956572558178, −12.14726399802373, −11.57611461470294, −11.12611593159678, −10.57482280866586, −10.02787626971215, −9.563989785328951, −9.108960819187496, −8.612420986032502, −8.009120281685951, −7.660952081733443, −7.235144260889674, −6.453529734283764, −6.195141604276382, −5.405240551567027, −4.734291781936116, −4.351255884530999, −3.669066711951937, −3.326340152827531, −2.468910329608097, −2.025512834498403, −1.342699929492535, −0.3216591470245596, 0.3216591470245596, 1.342699929492535, 2.025512834498403, 2.468910329608097, 3.326340152827531, 3.669066711951937, 4.351255884530999, 4.734291781936116, 5.405240551567027, 6.195141604276382, 6.453529734283764, 7.235144260889674, 7.660952081733443, 8.009120281685951, 8.612420986032502, 9.108960819187496, 9.563989785328951, 10.02787626971215, 10.57482280866586, 11.12611593159678, 11.57611461470294, 12.14726399802373, 12.50956572558178, 12.95863179582281, 13.58781678031827

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