Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17 $
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 − 4·11-s − 13-s − 17-s + 2·19-s + 2·21-s + 7·23-s − 27-s − 6·29-s + 4·31-s + 4·33-s − 11·37-s + 39-s − 9·41-s − 6·43-s − 4·47-s − 3·49-s + 51-s + 53-s − 2·57-s − 59-s + 5·61-s − 2·63-s + 2·67-s − 7·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.242·17-s + 0.458·19-s + 0.436·21-s + 1.45·23-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s − 1.80·37-s + 0.160·39-s − 1.40·41-s − 0.914·43-s − 0.583·47-s − 3/7·49-s + 0.140·51-s + 0.137·53-s − 0.264·57-s − 0.130·59-s + 0.640·61-s − 0.251·63-s + 0.244·67-s − 0.842·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(265200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{265200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 265200,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
13 \( 1 + T \)
17 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−12.89871951154582, −12.77949111491624, −12.04776115313502, −11.65690410539882, −11.21862253888881, −10.67920896574945, −10.26887251894502, −9.881153019272678, −9.488439279858390, −8.771944885388196, −8.468523476540744, −7.814790624712453, −7.258017601424743, −6.854120059123206, −6.575841429445975, −5.753188801619689, −5.402187152264206, −4.950422275099126, −4.577570761455581, −3.651477858151543, −3.240999854595865, −2.830024929718584, −2.016260819150337, −1.481331440178356, −0.5434060698479818, 0, 0.5434060698479818, 1.481331440178356, 2.016260819150337, 2.830024929718584, 3.240999854595865, 3.651477858151543, 4.577570761455581, 4.950422275099126, 5.402187152264206, 5.753188801619689, 6.575841429445975, 6.854120059123206, 7.258017601424743, 7.814790624712453, 8.468523476540744, 8.771944885388196, 9.488439279858390, 9.881153019272678, 10.26887251894502, 10.67920896574945, 11.21862253888881, 11.65690410539882, 12.04776115313502, 12.77949111491624, 12.89871951154582

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