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 + 6·13-s − 2·15-s − 2·17-s − 4·19-s − 25-s + 27-s + 2·29-s + 8·31-s − 33-s − 6·37-s + 6·39-s − 10·41-s − 4·43-s − 2·45-s − 8·47-s − 2·51-s − 6·53-s + 2·55-s − 4·57-s − 4·59-s − 10·61-s − 12·65-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.516·15-s − 0.485·17-s − 0.917·19-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.174·33-s − 0.986·37-s + 0.960·39-s − 1.56·41-s − 0.609·43-s − 0.298·45-s − 1.16·47-s − 0.280·51-s − 0.824·53-s + 0.269·55-s − 0.529·57-s − 0.520·59-s − 1.28·61-s − 1.48·65-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  =  $(2,\ 103488,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.244985955$
$L(\frac12)$  $\approx$  $1.244985955$
$L(\frac{3}{2})$   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 \( 1 - 6 T + p T^{2} \)
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

−13.69091381502370, −13.37159837966752, −12.78871783573565, −12.27055554345833, −11.73490295919921, −11.27112494438054, −10.85933179262444, −10.21047126081086, −9.942458744968332, −9.029496540137397, −8.593043034638283, −8.321321937366810, −7.949308285660674, −7.205909849042764, −6.677148188398076, −6.240064939915363, −5.653232505049799, −4.681471861709781, −4.456542779335204, −3.814263967702939, −3.178690617075192, −2.911672390630634, −1.760737574148101, −1.483347830642525, −0.3280024187788447, 0.3280024187788447, 1.483347830642525, 1.760737574148101, 2.911672390630634, 3.178690617075192, 3.814263967702939, 4.456542779335204, 4.681471861709781, 5.653232505049799, 6.240064939915363, 6.677148188398076, 7.205909849042764, 7.949308285660674, 8.321321937366810, 8.593043034638283, 9.029496540137397, 9.942458744968332, 10.21047126081086, 10.85933179262444, 11.27112494438054, 11.73490295919921, 12.27055554345833, 12.78871783573565, 13.37159837966752, 13.69091381502370

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