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 − 5·11-s − 13-s + 17-s + 19-s + 2·21-s − 4·23-s − 27-s − 4·29-s − 9·31-s + 5·33-s + 9·37-s + 39-s + 43-s − 6·47-s − 3·49-s − 51-s + 2·53-s − 57-s − 2·59-s − 9·61-s − 2·63-s − 12·67-s + 4·69-s + 6·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.242·17-s + 0.229·19-s + 0.436·21-s − 0.834·23-s − 0.192·27-s − 0.742·29-s − 1.61·31-s + 0.870·33-s + 1.47·37-s + 0.160·39-s + 0.152·43-s − 0.875·47-s − 3/7·49-s − 0.140·51-s + 0.274·53-s − 0.132·57-s − 0.260·59-s − 1.15·61-s − 0.251·63-s − 1.46·67-s + 0.481·69-s + 0.712·71-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  =  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 \( 1 + 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.95287859663926, −12.66395500861702, −12.12615383873778, −11.69533532921994, −11.10809507846586, −10.70100029967277, −10.34812336161020, −9.823995704324091, −9.369848184552442, −9.089964035451386, −8.101069328682163, −7.902296273101113, −7.417225301600932, −6.939381943836159, −6.305540946364902, −5.806163778514132, −5.528042467045804, −4.966917102494041, −4.390520383790407, −3.852067608464883, −3.136936926689895, −2.809170934861656, −2.039906193854720, −1.513147432957477, −0.4843857273189600, 0, 0.4843857273189600, 1.513147432957477, 2.039906193854720, 2.809170934861656, 3.136936926689895, 3.852067608464883, 4.390520383790407, 4.966917102494041, 5.528042467045804, 5.806163778514132, 6.305540946364902, 6.939381943836159, 7.417225301600932, 7.902296273101113, 8.101069328682163, 9.089964035451386, 9.369848184552442, 9.823995704324091, 10.34812336161020, 10.70100029967277, 11.10809507846586, 11.69533532921994, 12.12615383873778, 12.66395500861702, 12.95287859663926

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