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 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·7-s + 9-s − 3·11-s − 13-s + 17-s − 3·19-s + 3·21-s + 6·23-s − 27-s − 29-s + 8·31-s + 3·33-s − 7·37-s + 39-s − 3·41-s − 4·43-s + 5·47-s + 2·49-s − 51-s − 9·53-s + 3·57-s + 2·59-s − 12·61-s − 3·63-s − 14·67-s − 6·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 0.242·17-s − 0.688·19-s + 0.654·21-s + 1.25·23-s − 0.192·27-s − 0.185·29-s + 1.43·31-s + 0.522·33-s − 1.15·37-s + 0.160·39-s − 0.468·41-s − 0.609·43-s + 0.729·47-s + 2/7·49-s − 0.140·51-s − 1.23·53-s + 0.397·57-s + 0.260·59-s − 1.53·61-s − 0.377·63-s − 1.71·67-s − 0.722·69-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  =  2
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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−13.22356199121630, −12.70568103364930, −12.43375580688123, −11.93126333577689, −11.48137773675238, −10.81197450637617, −10.39297860709770, −10.27285306571803, −9.582764550961804, −9.168110744545788, −8.681752321601924, −8.077739053758757, −7.567495189475234, −7.092485299099157, −6.524570137604783, −6.287089754031250, −5.685754525393839, −5.040052675536338, −4.820918939810049, −4.119033684506090, −3.448410670228364, −2.924429640638466, −2.589456754065554, −1.682765828523448, −1.060391405858335, 0, 0, 1.060391405858335, 1.682765828523448, 2.589456754065554, 2.924429640638466, 3.448410670228364, 4.119033684506090, 4.820918939810049, 5.040052675536338, 5.685754525393839, 6.287089754031250, 6.524570137604783, 7.092485299099157, 7.567495189475234, 8.077739053758757, 8.681752321601924, 9.168110744545788, 9.582764550961804, 10.27285306571803, 10.39297860709770, 10.81197450637617, 11.48137773675238, 11.93126333577689, 12.43375580688123, 12.70568103364930, 13.22356199121630

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