Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s + 11-s + 4·13-s − 4·19-s − 6·23-s + 6·29-s + 8·31-s − 2·37-s − 6·41-s − 8·43-s + 6·47-s + 9·49-s − 6·53-s + 12·59-s + 2·61-s + 10·67-s + 12·71-s + 16·73-s + 4·77-s + 8·79-s − 6·89-s + 16·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 1.51·7-s + 0.301·11-s + 1.10·13-s − 0.917·19-s − 1.25·23-s + 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.937·41-s − 1.21·43-s + 0.875·47-s + 9/7·49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s + 1.22·67-s + 1.42·71-s + 1.87·73-s + 0.455·77-s + 0.900·79-s − 0.635·89-s + 1.67·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{9900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 9900,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.854447782\)
\(L(\frac12)\)  \(\approx\)  \(2.854447782\)
\(L(\frac{3}{2})\)   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 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−16.81715682949398, −15.93547865541052, −15.52979211175083, −14.94095855914610, −14.28176145492909, −13.83066829427583, −13.43659078658385, −12.40391517860489, −12.01170499208075, −11.35280219802113, −10.90824482019974, −10.26476629455678, −9.648778391951944, −8.626905073857228, −8.176840143993500, −8.097597377267241, −6.760553725009778, −6.481069683701251, −5.533357169406400, −4.891734854529347, −4.199504690435894, −3.601239384372207, −2.398673650281497, −1.713684078850284, −0.8469248868911995, 0.8469248868911995, 1.713684078850284, 2.398673650281497, 3.601239384372207, 4.199504690435894, 4.891734854529347, 5.533357169406400, 6.481069683701251, 6.760553725009778, 8.097597377267241, 8.176840143993500, 8.626905073857228, 9.648778391951944, 10.26476629455678, 10.90824482019974, 11.35280219802113, 12.01170499208075, 12.40391517860489, 13.43659078658385, 13.83066829427583, 14.28176145492909, 14.94095855914610, 15.52979211175083, 15.93547865541052, 16.81715682949398

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