Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7^{2} \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s − 11-s + 12-s + 6·13-s − 16-s + 2·17-s + 18-s − 4·19-s − 22-s + 3·24-s + 6·26-s − 27-s − 2·29-s − 8·31-s + 5·32-s + 33-s + 2·34-s − 36-s − 6·37-s − 4·38-s − 6·39-s − 10·41-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 1.66·13-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.213·22-s + 0.612·24-s + 1.17·26-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.883·32-s + 0.174·33-s + 0.342·34-s − 1/6·36-s − 0.986·37-s − 0.648·38-s − 0.960·39-s − 1.56·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(40425\)    =    \(3 \cdot 5^{2} \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{40425} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 40425,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.336592331$
$L(\frac12)$  $\approx$  $1.336592331$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−14.78398705674851, −14.10895579422127, −13.71928687172613, −13.13074778018838, −12.79122503345408, −12.36135409707505, −11.69287516533205, −11.10343254711680, −10.77773375555751, −10.10805198480978, −9.472128111641309, −8.950933510829235, −8.281289530877027, −8.030626937979085, −6.886485681104650, −6.598542413173404, −5.828071917915865, −5.491402351492877, −4.932027633271936, −4.231696959593402, −3.525760105863025, −3.388625988053637, −2.143297775001162, −1.400999528453565, −0.4031707294308967, 0.4031707294308967, 1.400999528453565, 2.143297775001162, 3.388625988053637, 3.525760105863025, 4.231696959593402, 4.932027633271936, 5.491402351492877, 5.828071917915865, 6.598542413173404, 6.886485681104650, 8.030626937979085, 8.281289530877027, 8.950933510829235, 9.472128111641309, 10.10805198480978, 10.77773375555751, 11.10343254711680, 11.69287516533205, 12.36135409707505, 12.79122503345408, 13.13074778018838, 13.71928687172613, 14.10895579422127, 14.78398705674851

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