Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 11 \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 − 2·5-s + 2·7-s + 9-s − 11-s − 2·13-s − 2·15-s − 4·17-s + 6·19-s + 2·21-s − 25-s + 27-s − 8·29-s − 8·31-s − 33-s − 4·35-s − 10·37-s − 2·39-s + 8·41-s + 2·43-s − 2·45-s − 8·47-s − 3·49-s − 4·51-s + 2·53-s + 2·55-s + 6·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.516·15-s − 0.970·17-s + 1.37·19-s + 0.436·21-s − 1/5·25-s + 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.174·33-s − 0.676·35-s − 1.64·37-s − 0.320·39-s + 1.24·41-s + 0.304·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s + 0.269·55-s + 0.794·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(69828\)    =    \(2^{2} \cdot 3 \cdot 11 \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{69828} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 69828,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.267867567$
$L(\frac12)$  $\approx$  $1.267867567$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
11 \( 1 + T \)
23 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−14.21494420826709, −13.69038883807000, −13.16579650390277, −12.67082309353978, −12.09558968207576, −11.62313639058311, −11.06535330851354, −10.88033118014188, −9.991842113690428, −9.505442068557546, −8.994599355209353, −8.518896048622961, −7.852753096352598, −7.456907824280535, −7.266995886052320, −6.464926353682151, −5.515636635736535, −5.204341688252581, −4.536508625714030, −3.897563028220118, −3.492498412413531, −2.760116103768881, −1.992843150511161, −1.510728223575089, −0.3480911112435095, 0.3480911112435095, 1.510728223575089, 1.992843150511161, 2.760116103768881, 3.492498412413531, 3.897563028220118, 4.536508625714030, 5.204341688252581, 5.515636635736535, 6.464926353682151, 7.266995886052320, 7.456907824280535, 7.852753096352598, 8.518896048622961, 8.994599355209353, 9.505442068557546, 9.991842113690428, 10.88033118014188, 11.06535330851354, 11.62313639058311, 12.09558968207576, 12.67082309353978, 13.16579650390277, 13.69038883807000, 14.21494420826709

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