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 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s + 9-s − 5·11-s − 13-s − 17-s + 7·19-s + 4·21-s − 6·23-s − 27-s + 4·29-s + 9·31-s + 5·33-s − 5·37-s + 39-s − 2·41-s + 43-s − 12·47-s + 9·49-s + 51-s − 7·57-s − 4·59-s + 11·61-s − 4·63-s − 10·67-s + 6·69-s + 14·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 0.242·17-s + 1.60·19-s + 0.872·21-s − 1.25·23-s − 0.192·27-s + 0.742·29-s + 1.61·31-s + 0.870·33-s − 0.821·37-s + 0.160·39-s − 0.312·41-s + 0.152·43-s − 1.75·47-s + 9/7·49-s + 0.140·51-s − 0.927·57-s − 0.520·59-s + 1.40·61-s − 0.503·63-s − 1.22·67-s + 0.722·69-s + 1.66·71-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  =  0
Selberg data  =  $(2,\ 265200,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6967923391$
$L(\frac12)$  $\approx$  $0.6967923391$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−12.77613874500216, −12.33301902915740, −11.92158678730597, −11.54737575156571, −10.91149603474086, −10.29071762921392, −10.03872029828652, −9.776576866592645, −9.332791227574899, −8.381519608772940, −8.250519507196841, −7.530835795683761, −7.121128077013740, −6.582448612825585, −6.189097988900124, −5.663748091204157, −5.181439126259493, −4.751432030640318, −4.119417295387662, −3.318735439446702, −3.079620351546805, −2.509245981534767, −1.814243720822885, −0.8764974004468950, −0.2895593487033456, 0.2895593487033456, 0.8764974004468950, 1.814243720822885, 2.509245981534767, 3.079620351546805, 3.318735439446702, 4.119417295387662, 4.751432030640318, 5.181439126259493, 5.663748091204157, 6.189097988900124, 6.582448612825585, 7.121128077013740, 7.530835795683761, 8.250519507196841, 8.381519608772940, 9.332791227574899, 9.776576866592645, 10.03872029828652, 10.29071762921392, 10.91149603474086, 11.54737575156571, 11.92158678730597, 12.33301902915740, 12.77613874500216

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