Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 2·13-s − 4·17-s + 6·19-s − 8·29-s − 8·31-s − 10·37-s + 8·41-s − 2·43-s − 8·47-s − 3·49-s − 2·53-s − 12·59-s − 10·61-s − 12·67-s − 8·71-s + 6·73-s + 2·79-s − 16·83-s + 14·89-s + 4·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.755·7-s − 0.554·13-s − 0.970·17-s + 1.37·19-s − 1.48·29-s − 1.43·31-s − 1.64·37-s + 1.24·41-s − 0.304·43-s − 1.16·47-s − 3/7·49-s − 0.274·53-s − 1.56·59-s − 1.28·61-s − 1.46·67-s − 0.949·71-s + 0.702·73-s + 0.225·79-s − 1.75·83-s + 1.48·89-s + 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{108900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 108900,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−14.21790960796929, −13.47657173641210, −13.27832836940350, −12.70838530493340, −12.23171255127185, −11.81281155743730, −11.07661224305758, −10.89423304044256, −10.21222219302713, −9.546148230983968, −9.336214860342957, −8.935943683034039, −8.177290160048833, −7.471054936725398, −7.275625707240563, −6.718553008471583, −5.961802478973440, −5.689117468615944, −4.904508505906366, −4.543354942397739, −3.599653950673350, −3.352957381447465, −2.685858041692292, −1.873371449557278, −1.387987268460352, 0, 0, 1.387987268460352, 1.873371449557278, 2.685858041692292, 3.352957381447465, 3.599653950673350, 4.543354942397739, 4.904508505906366, 5.689117468615944, 5.961802478973440, 6.718553008471583, 7.275625707240563, 7.471054936725398, 8.177290160048833, 8.935943683034039, 9.336214860342957, 9.546148230983968, 10.21222219302713, 10.89423304044256, 11.07661224305758, 11.81281155743730, 12.23171255127185, 12.70838530493340, 13.27832836940350, 13.47657173641210, 14.21790960796929

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