Properties

Degree 2
Conductor $ 2 \cdot 3 \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 − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s − 12-s − 4·13-s − 2·14-s + 16-s − 6·17-s − 18-s + 4·19-s − 2·21-s − 6·23-s + 24-s + 4·26-s − 27-s + 2·28-s − 6·29-s + 8·31-s − 32-s + 6·34-s + 36-s + 10·37-s − 4·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.436·21-s − 1.25·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.377·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.64·37-s − 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(18150\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{18150} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 18150,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.8433000708\)
\(L(\frac12)\)  \(\approx\)  \(0.8433000708\)
\(L(\frac{3}{2})\)   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 \( 1 + T \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−15.83921548228591, −15.38747587225585, −14.84768867925781, −14.17970097337979, −13.64548936037945, −12.95017606133921, −12.21815841715427, −11.84840531346874, −11.21967643697253, −10.93531579807689, −10.09645847043574, −9.650612194534090, −9.172686227873408, −8.252304364355296, −7.905591873485777, −7.214506467005306, −6.722345246158730, −5.900799202432833, −5.416075755736886, −4.478705838911386, −4.218694145481117, −2.889675108848370, −2.239767330633625, −1.483764006939233, −0.4509071286736605, 0.4509071286736605, 1.483764006939233, 2.239767330633625, 2.889675108848370, 4.218694145481117, 4.478705838911386, 5.416075755736886, 5.900799202432833, 6.722345246158730, 7.214506467005306, 7.905591873485777, 8.252304364355296, 9.172686227873408, 9.650612194534090, 10.09645847043574, 10.93531579807689, 11.21967643697253, 11.84840531346874, 12.21815841715427, 12.95017606133921, 13.64548936037945, 14.17970097337979, 14.84768867925781, 15.38747587225585, 15.83921548228591

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