Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 4·7-s + 8-s + 2·13-s − 4·14-s + 16-s − 6·17-s + 4·19-s + 2·26-s − 4·28-s − 6·29-s + 8·31-s + 32-s − 6·34-s − 2·37-s + 4·38-s − 6·41-s − 4·43-s + 9·49-s + 2·52-s − 6·53-s − 4·56-s − 6·58-s + 10·61-s + 8·62-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.392·26-s − 0.755·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.937·41-s − 0.609·43-s + 9/7·49-s + 0.277·52-s − 0.824·53-s − 0.534·56-s − 0.787·58-s + 1.28·61-s + 1.01·62-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(54450\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{54450} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 54450,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.087590544\)
\(L(\frac12)\)  \(\approx\)  \(2.087590544\)
\(L(\frac{3}{2})\)   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 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−14.22973391649383, −13.83436679518390, −13.26256621956431, −13.08749893030217, −12.54862163211068, −11.92786200941264, −11.35417386653792, −11.07561768572633, −10.17177625698024, −9.903911172926840, −9.354330858459932, −8.645945422891212, −8.274674249644647, −7.239090051164330, −7.004395228747958, −6.353040722691544, −6.027707569264893, −5.325053568434956, −4.684243289881667, −4.012007592523062, −3.451957191837842, −2.985272286385384, −2.303186119853407, −1.475718123936882, −0.4328467891184769, 0.4328467891184769, 1.475718123936882, 2.303186119853407, 2.985272286385384, 3.451957191837842, 4.012007592523062, 4.684243289881667, 5.325053568434956, 6.027707569264893, 6.353040722691544, 7.004395228747958, 7.239090051164330, 8.274674249644647, 8.645945422891212, 9.354330858459932, 9.903911172926840, 10.17177625698024, 11.07561768572633, 11.35417386653792, 11.92786200941264, 12.54862163211068, 13.08749893030217, 13.26256621956431, 13.83436679518390, 14.22973391649383

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