Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 23^{2} $
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 − 2·4-s + 5-s − 7-s + 9-s + 4·11-s − 2·12-s + 3·13-s + 15-s + 4·16-s − 17-s − 2·20-s − 21-s + 25-s + 27-s + 2·28-s − 3·29-s − 6·31-s + 4·33-s − 35-s − 2·36-s − 6·37-s + 3·39-s + 12·41-s + 6·43-s − 8·44-s + 45-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 0.832·13-s + 0.258·15-s + 16-s − 0.242·17-s − 0.447·20-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 0.557·29-s − 1.07·31-s + 0.696·33-s − 0.169·35-s − 1/3·36-s − 0.986·37-s + 0.480·39-s + 1.87·41-s + 0.914·43-s − 1.20·44-s + 0.149·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(55545\)    =    \(3 \cdot 5 \cdot 7 \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{55545} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 55545,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.973054025$
$L(\frac12)$  $\approx$  $2.973054025$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−14.28058591110356, −14.05862443690149, −13.25614882317738, −13.16793387266413, −12.42289288545138, −12.12718531381707, −11.15829635883685, −10.84024492672904, −10.12370264600170, −9.552545387186451, −9.206706127720522, −8.772648418863909, −8.460702718147605, −7.538942371013228, −7.137295084520491, −6.439555997809479, −5.668371402798628, −5.567578331508354, −4.426881491943636, −4.022876700530663, −3.658104279313566, −2.881961875147359, −2.060776467029293, −1.298900537836478, −0.6366784893316967, 0.6366784893316967, 1.298900537836478, 2.060776467029293, 2.881961875147359, 3.658104279313566, 4.022876700530663, 4.426881491943636, 5.567578331508354, 5.668371402798628, 6.439555997809479, 7.137295084520491, 7.538942371013228, 8.460702718147605, 8.772648418863909, 9.206706127720522, 9.552545387186451, 10.12370264600170, 10.84024492672904, 11.15829635883685, 12.12718531381707, 12.42289288545138, 13.16793387266413, 13.25614882317738, 14.05862443690149, 14.28058591110356

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