Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5 \cdot 7 $
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 − 5-s − 7-s + 9-s + 4·11-s + 2·13-s + 15-s − 6·17-s + 4·19-s + 21-s − 8·23-s + 25-s − 27-s + 2·29-s − 4·33-s + 35-s + 2·37-s − 2·39-s + 10·41-s + 4·43-s − 45-s + 49-s + 6·51-s − 14·53-s − 4·55-s − 4·57-s + 12·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s + 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.696·33-s + 0.169·35-s + 0.328·37-s − 0.320·39-s + 1.56·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.840·51-s − 1.92·53-s − 0.539·55-s − 0.529·57-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6720\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6720} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6720,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.296562659$
$L(\frac12)$  $\approx$  $1.296562659$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−17.39255902225985, −16.50943526042553, −16.00350202608555, −15.79049589037659, −14.96859590924547, −14.15401373236236, −13.81962809695233, −12.93296607702572, −12.43526064209967, −11.71750942460115, −11.34601117932484, −10.76586893431916, −9.883624743610455, −9.354252007175672, −8.709358387415246, −7.914672562000366, −7.205759239351276, −6.353832018086827, −6.179477732472510, −5.143034152606122, −4.108124084663131, −3.969247713400847, −2.770559863397882, −1.669658176382584, −0.6164088564067552, 0.6164088564067552, 1.669658176382584, 2.770559863397882, 3.969247713400847, 4.108124084663131, 5.143034152606122, 6.179477732472510, 6.353832018086827, 7.205759239351276, 7.914672562000366, 8.709358387415246, 9.354252007175672, 9.883624743610455, 10.76586893431916, 11.34601117932484, 11.71750942460115, 12.43526064209967, 12.93296607702572, 13.81962809695233, 14.15401373236236, 14.96859590924547, 15.79049589037659, 16.00350202608555, 16.50943526042553, 17.39255902225985

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