Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 13^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 4·11-s − 15-s + 2·17-s + 4·19-s + 25-s + 27-s − 2·29-s − 4·33-s + 10·37-s − 10·41-s − 4·43-s − 45-s + 8·47-s − 7·49-s + 2·51-s − 10·53-s + 4·55-s + 4·57-s − 4·59-s − 2·61-s + 12·67-s − 8·71-s − 10·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.258·15-s + 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.696·33-s + 1.64·37-s − 1.56·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.280·51-s − 1.37·53-s + 0.539·55-s + 0.529·57-s − 0.520·59-s − 0.256·61-s + 1.46·67-s − 0.949·71-s − 1.17·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(40560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{40560} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 40560,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
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 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.91435929626356, −14.65792821341048, −13.93300840344889, −13.48978105186497, −12.99346528732752, −12.55131058131567, −11.86841291003575, −11.43744845556903, −10.79015863197572, −10.27211776612621, −9.698681570988576, −9.307667824193484, −8.459222658755652, −8.123480794848969, −7.508280369094317, −7.273514065172921, −6.332490846188990, −5.777216437279961, −4.945828276221568, −4.706005268589482, −3.688843022456496, −3.246161042625413, −2.666557436388239, −1.879506903155439, −0.9995151759850184, 0, 0.9995151759850184, 1.879506903155439, 2.666557436388239, 3.246161042625413, 3.688843022456496, 4.706005268589482, 4.945828276221568, 5.777216437279961, 6.332490846188990, 7.273514065172921, 7.508280369094317, 8.123480794848969, 8.459222658755652, 9.307667824193484, 9.698681570988576, 10.27211776612621, 10.79015863197572, 11.43744845556903, 11.86841291003575, 12.55131058131567, 12.99346528732752, 13.48978105186497, 13.93300840344889, 14.65792821341048, 14.91435929626356

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