Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \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  − 3-s + 2·5-s + 9-s + 11-s − 2·13-s − 2·15-s − 4·17-s + 6·19-s − 25-s − 27-s + 8·29-s − 8·31-s − 33-s − 10·37-s + 2·39-s − 8·41-s − 2·43-s + 2·45-s − 8·47-s + 4·51-s + 2·53-s + 2·55-s − 6·57-s − 12·59-s + 10·61-s − 4·65-s + 12·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-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 − 1/5·25-s − 0.192·27-s + 1.48·29-s − 1.43·31-s − 0.174·33-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 + 0.560·51-s + 0.274·53-s + 0.269·55-s − 0.794·57-s − 1.56·59-s + 1.28·61-s − 0.496·65-s + 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(103488\)    =    \(2^{6} \cdot 3 \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{103488} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 103488,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.419867659$
$L(\frac12)$  $\approx$  $1.419867659$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
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

−13.77490799610617, −13.16911630619210, −12.86479411410299, −12.04833317188193, −11.90157217976739, −11.32599381783658, −10.72950848330165, −10.26181507232395, −9.744860054505169, −9.501663096903929, −8.739875338843922, −8.415666776411920, −7.551038695668430, −7.028378557674495, −6.702526246601968, −6.088628816248917, −5.540705018566047, −5.012898437947142, −4.730712962291209, −3.805868738635795, −3.277478212456016, −2.538830043066465, −1.786004928359049, −1.424581012331155, −0.3762381646149590, 0.3762381646149590, 1.424581012331155, 1.786004928359049, 2.538830043066465, 3.277478212456016, 3.805868738635795, 4.730712962291209, 5.012898437947142, 5.540705018566047, 6.088628816248917, 6.702526246601968, 7.028378557674495, 7.551038695668430, 8.415666776411920, 8.739875338843922, 9.501663096903929, 9.744860054505169, 10.26181507232395, 10.72950848330165, 11.32599381783658, 11.90157217976739, 12.04833317188193, 12.86479411410299, 13.16911630619210, 13.77490799610617

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