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 − 2·7-s + 9-s − 3·11-s + 13-s − 17-s − 2·19-s + 2·21-s − 27-s − 6·29-s + 4·31-s + 3·33-s − 4·37-s − 39-s + 9·41-s + 43-s − 3·47-s − 3·49-s + 51-s + 6·53-s + 2·57-s − 12·59-s − 7·61-s − 2·63-s − 8·67-s + 6·71-s − 7·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s − 0.242·17-s − 0.458·19-s + 0.436·21-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.522·33-s − 0.657·37-s − 0.160·39-s + 1.40·41-s + 0.152·43-s − 0.437·47-s − 3/7·49-s + 0.140·51-s + 0.824·53-s + 0.264·57-s − 1.56·59-s − 0.896·61-s − 0.251·63-s − 0.977·67-s + 0.712·71-s − 0.819·73-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−13.29474227008221, −12.75854274585254, −12.35926689900694, −11.99864069465731, −11.27883467539882, −10.95288238070932, −10.57146028951462, −10.08684772709245, −9.621857157561711, −9.122175316977675, −8.716492500833943, −7.959032634703011, −7.709501074685032, −7.056964207410390, −6.650872309967785, −5.996050690174664, −5.846865536244526, −5.163922144475716, −4.645962831592606, −4.125591648229248, −3.570425925267236, −2.903003891432085, −2.494234392561408, −1.711641528691207, −1.086806036524822, 0, 0, 1.086806036524822, 1.711641528691207, 2.494234392561408, 2.903003891432085, 3.570425925267236, 4.125591648229248, 4.645962831592606, 5.163922144475716, 5.846865536244526, 5.996050690174664, 6.650872309967785, 7.056964207410390, 7.709501074685032, 7.959032634703011, 8.716492500833943, 9.122175316977675, 9.621857157561711, 10.08684772709245, 10.57146028951462, 10.95288238070932, 11.27883467539882, 11.99864069465731, 12.35926689900694, 12.75854274585254, 13.29474227008221

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