Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \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  + 5-s + 4·7-s + 4·11-s − 2·17-s − 4·19-s − 4·23-s + 25-s + 2·29-s + 8·31-s + 4·35-s − 6·37-s − 6·41-s − 8·43-s + 4·47-s + 9·49-s − 6·53-s + 4·55-s − 4·59-s − 2·61-s − 8·67-s + 6·73-s + 16·77-s − 16·83-s − 2·85-s − 6·89-s − 4·95-s + 14·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s + 1.20·11-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.676·35-s − 0.986·37-s − 0.937·41-s − 1.21·43-s + 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.539·55-s − 0.520·59-s − 0.256·61-s − 0.977·67-s + 0.702·73-s + 1.82·77-s − 1.75·83-s − 0.216·85-s − 0.635·89-s − 0.410·95-s + 1.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(60840\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{60840} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 60840,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.40394558155344, −14.17769441327096, −13.61284134840028, −13.22509962175794, −12.27760258523049, −12.03011679557568, −11.57286512619729, −11.01889978517044, −10.48398977549484, −10.04720403618776, −9.380971309113663, −8.733690891566053, −8.428421302770586, −7.971909759711102, −7.219959947943595, −6.506235401288905, −6.338658362761998, −5.473315934621432, −4.893570059691219, −4.393910228624320, −3.947324885885618, −3.049520617352223, −2.237146356751511, −1.635618359399914, −1.247612373184159, 0, 1.247612373184159, 1.635618359399914, 2.237146356751511, 3.049520617352223, 3.947324885885618, 4.393910228624320, 4.893570059691219, 5.473315934621432, 6.338658362761998, 6.506235401288905, 7.219959947943595, 7.971909759711102, 8.428421302770586, 8.733690891566053, 9.380971309113663, 10.04720403618776, 10.48398977549484, 11.01889978517044, 11.57286512619729, 12.03011679557568, 12.27760258523049, 13.22509962175794, 13.61284134840028, 14.17769441327096, 14.40394558155344

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