Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·7-s + 9-s + 11-s + 4·13-s − 17-s + 6·19-s − 2·21-s − 2·23-s + 27-s − 6·29-s + 4·31-s + 33-s + 2·37-s + 4·39-s − 6·41-s + 6·43-s − 12·47-s − 3·49-s − 51-s + 6·57-s − 6·59-s + 2·61-s − 2·63-s − 2·69-s − 6·71-s + 4·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.242·17-s + 1.37·19-s − 0.436·21-s − 0.417·23-s + 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.174·33-s + 0.328·37-s + 0.640·39-s − 0.937·41-s + 0.914·43-s − 1.75·47-s − 3/7·49-s − 0.140·51-s + 0.794·57-s − 0.781·59-s + 0.256·61-s − 0.251·63-s − 0.240·69-s − 0.712·71-s + 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{224400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 224400,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.460430843$
$L(\frac12)$  $\approx$  $2.460430843$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
11 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−13.18565796240978, −12.53242054921401, −12.05391447955852, −11.53547637765776, −11.14406828303957, −10.62960727785698, −9.940953555511979, −9.669724793576909, −9.266872401559634, −8.773520602885108, −8.183276925817710, −7.878303542985447, −7.237481199062236, −6.722668593996340, −6.332679616792947, −5.766050753105152, −5.283935190519453, −4.572902394619332, −3.959278932605949, −3.574843302519485, −3.033641608144472, −2.628495917760087, −1.606338101221389, −1.385772507855635, −0.4151151767981522, 0.4151151767981522, 1.385772507855635, 1.606338101221389, 2.628495917760087, 3.033641608144472, 3.574843302519485, 3.959278932605949, 4.572902394619332, 5.283935190519453, 5.766050753105152, 6.332679616792947, 6.722668593996340, 7.237481199062236, 7.878303542985447, 8.183276925817710, 8.773520602885108, 9.266872401559634, 9.669724793576909, 9.940953555511979, 10.62960727785698, 11.14406828303957, 11.53547637765776, 12.05391447955852, 12.53242054921401, 13.18565796240978

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