Properties

Degree 2
Conductor $ 3^{2} \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  − 2-s − 4-s − 5-s + 3·8-s + 10-s + 2·13-s − 16-s + 2·17-s − 4·19-s + 20-s + 25-s − 2·26-s − 2·29-s − 5·32-s − 2·34-s − 10·37-s + 4·38-s − 3·40-s + 10·41-s − 4·43-s − 8·47-s − 7·49-s − 50-s − 2·52-s + 10·53-s + 2·58-s + 4·59-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s + 0.316·10-s + 0.554·13-s − 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s − 0.392·26-s − 0.371·29-s − 0.883·32-s − 0.342·34-s − 1.64·37-s + 0.648·38-s − 0.474·40-s + 1.56·41-s − 0.609·43-s − 1.16·47-s − 49-s − 0.141·50-s − 0.277·52-s + 1.37·53-s + 0.262·58-s + 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5445\)    =    \(3^{2} \cdot 5 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{5445} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 5445,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
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

−17.90485967374580, −17.37625581540748, −16.79089322917527, −16.16811417070434, −15.73071124979747, −14.71456427051100, −14.48864917053516, −13.53836079282137, −13.11597719023717, −12.43419214500324, −11.71692174461908, −10.90260394159546, −10.52872269551793, −9.722000487307416, −9.173760699770040, −8.327185665547056, −8.168912727661557, −7.228968944427759, −6.583727948418504, −5.577922906944658, −4.873494082101879, −4.014466479236401, −3.450009603472106, −2.131438235194789, −1.110503936373346, 0, 1.110503936373346, 2.131438235194789, 3.450009603472106, 4.014466479236401, 4.873494082101879, 5.577922906944658, 6.583727948418504, 7.228968944427759, 8.168912727661557, 8.327185665547056, 9.173760699770040, 9.722000487307416, 10.52872269551793, 10.90260394159546, 11.71692174461908, 12.43419214500324, 13.11597719023717, 13.53836079282137, 14.48864917053516, 14.71456427051100, 15.73071124979747, 16.16811417070434, 16.79089322917527, 17.37625581540748, 17.90485967374580

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