Properties

Degree 2
Conductor $ 2^{4} \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  + 2·3-s + 5-s + 2·7-s + 9-s + 2·15-s − 6·17-s − 4·19-s + 4·21-s − 6·23-s + 25-s − 4·27-s + 6·29-s − 4·31-s + 2·35-s − 2·37-s − 6·41-s + 10·43-s + 45-s − 6·47-s − 3·49-s − 12·51-s − 6·53-s − 8·57-s + 12·59-s + 2·61-s + 2·63-s + 2·67-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.516·15-s − 1.45·17-s − 0.917·19-s + 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s + 0.338·35-s − 0.328·37-s − 0.937·41-s + 1.52·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 1.68·51-s − 0.824·53-s − 1.05·57-s + 1.56·59-s + 0.256·61-s + 0.251·63-s + 0.244·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13520\)    =    \(2^{4} \cdot 5 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{13520} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 13520,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
13 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−16.29945794667661, −15.85473543262656, −15.22897757811834, −14.60836073061809, −14.33525549921493, −13.79156091335017, −13.19991219149557, −12.79431241770244, −11.89697000155386, −11.37562108279264, −10.68690660531323, −10.16820731395752, −9.409212641265711, −8.875851008205262, −8.358953963528495, −8.012352711093259, −7.148998896521483, −6.486954586170532, −5.836479677528695, −4.940491038559921, −4.310778491711975, −3.676860164500569, −2.681710179007272, −2.161331153306945, −1.559052445577068, 0, 1.559052445577068, 2.161331153306945, 2.681710179007272, 3.676860164500569, 4.310778491711975, 4.940491038559921, 5.836479677528695, 6.486954586170532, 7.148998896521483, 8.012352711093259, 8.358953963528495, 8.875851008205262, 9.409212641265711, 10.16820731395752, 10.68690660531323, 11.37562108279264, 11.89697000155386, 12.79431241770244, 13.19991219149557, 13.79156091335017, 14.33525549921493, 14.60836073061809, 15.22897757811834, 15.85473543262656, 16.29945794667661

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