Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s + 9-s − 11-s − 6·13-s − 6·17-s − 8·19-s + 4·21-s − 27-s − 6·29-s + 33-s − 6·37-s + 6·39-s − 10·41-s + 8·43-s + 9·49-s + 6·51-s − 6·53-s + 8·57-s + 4·59-s − 2·61-s − 4·63-s + 12·67-s − 8·71-s − 2·73-s + 4·77-s − 4·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s − 1.45·17-s − 1.83·19-s + 0.872·21-s − 0.192·27-s − 1.11·29-s + 0.174·33-s − 0.986·37-s + 0.960·39-s − 1.56·41-s + 1.21·43-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 1.05·57-s + 0.520·59-s − 0.256·61-s − 0.503·63-s + 1.46·67-s − 0.949·71-s − 0.234·73-s + 0.455·77-s − 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6600\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6600} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((2,\ 6600,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
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

−6.91243348572659873459197337224, −6.77114453439511230036963943917, −5.98633093606997687788710606516, −5.15844403752174915099379488888, −4.43574802067698234038978674358, −3.68567392097867542340724775466, −2.62166115943425452399518566435, −2.01172862456584052085901183106, 0, 0, 2.01172862456584052085901183106, 2.62166115943425452399518566435, 3.68567392097867542340724775466, 4.43574802067698234038978674358, 5.15844403752174915099379488888, 5.98633093606997687788710606516, 6.77114453439511230036963943917, 6.91243348572659873459197337224

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