Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5^{2} \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 − 2·7-s + 9-s − 2·13-s + 4·17-s + 6·19-s + 2·21-s − 27-s + 8·29-s − 8·31-s − 10·37-s + 2·39-s − 8·41-s − 2·43-s + 8·47-s − 3·49-s − 4·51-s + 2·53-s − 6·57-s + 12·59-s − 10·61-s − 2·63-s − 12·67-s + 8·71-s + 6·73-s + 2·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.554·13-s + 0.970·17-s + 1.37·19-s + 0.436·21-s − 0.192·27-s + 1.48·29-s − 1.43·31-s − 1.64·37-s + 0.320·39-s − 1.24·41-s − 0.304·43-s + 1.16·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s − 0.794·57-s + 1.56·59-s − 1.28·61-s − 0.251·63-s − 1.46·67-s + 0.949·71-s + 0.702·73-s + 0.225·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36300\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{36300} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 36300,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 \)
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

−15.31970280728218, −14.61206007985636, −14.01014541618542, −13.68266400803999, −12.99882098774423, −12.34785758417019, −12.09736572502415, −11.67580517537190, −10.87815851902324, −10.26656687474574, −10.01874102265998, −9.386864776157856, −8.851606899279576, −8.115660212188827, −7.390991461700994, −7.076464797162382, −6.431268114970880, −5.786328049745563, −5.156750674967987, −4.892004684575666, −3.739587869258921, −3.407162494430413, −2.668122776055509, −1.702225038665370, −0.9049959723921419, 0, 0.9049959723921419, 1.702225038665370, 2.668122776055509, 3.407162494430413, 3.739587869258921, 4.892004684575666, 5.156750674967987, 5.786328049745563, 6.431268114970880, 7.076464797162382, 7.390991461700994, 8.115660212188827, 8.851606899279576, 9.386864776157856, 10.01874102265998, 10.26656687474574, 10.87815851902324, 11.67580517537190, 12.09736572502415, 12.34785758417019, 12.99882098774423, 13.68266400803999, 14.01014541618542, 14.61206007985636, 15.31970280728218

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