Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 11^{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 + 2·13-s + 15-s − 2·17-s + 4·19-s + 25-s + 27-s + 2·29-s − 10·37-s + 2·39-s − 10·41-s + 4·43-s + 45-s − 8·47-s − 7·49-s − 2·51-s − 10·53-s + 4·57-s + 4·59-s + 2·61-s + 2·65-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 + 0.554·13-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 − 1.64·37-s + 0.320·39-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 + 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.248·65-s − 1.46·67-s + 0.949·71-s − 1.17·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29040\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{29040} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 29040,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 \)
good7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
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

−15.45321514062981, −14.88476678722055, −14.25108989409567, −13.90495202074318, −13.35928539951467, −12.98758520531127, −12.26515854730900, −11.73385704390607, −11.14858786303303, −10.48541506539559, −10.03920193380913, −9.429311114853201, −8.933614044804268, −8.385135639329550, −7.869765249767162, −7.129221463097211, −6.594977055346115, −6.071097909454496, −5.149555251941368, −4.861582671100601, −3.865874751857574, −3.323959007578207, −2.720844950966845, −1.783181842092050, −1.312943511969394, 0, 1.312943511969394, 1.783181842092050, 2.720844950966845, 3.323959007578207, 3.865874751857574, 4.861582671100601, 5.149555251941368, 6.071097909454496, 6.594977055346115, 7.129221463097211, 7.869765249767162, 8.385135639329550, 8.933614044804268, 9.429311114853201, 10.03920193380913, 10.48541506539559, 11.14858786303303, 11.73385704390607, 12.26515854730900, 12.98758520531127, 13.35928539951467, 13.90495202074318, 14.25108989409567, 14.88476678722055, 15.45321514062981

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