Properties

Degree 2
Conductor $ 2^{4} \cdot 5 \cdot 29^{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  + 5-s + 4·7-s − 3·9-s + 4·11-s − 2·13-s − 2·17-s + 4·19-s − 4·23-s + 25-s − 8·31-s + 4·35-s − 6·37-s + 6·41-s − 8·43-s − 3·45-s + 4·47-s + 9·49-s + 6·53-s + 4·55-s + 4·59-s + 2·61-s − 12·63-s − 2·65-s − 8·67-s + 6·73-s + 16·77-s + 9·81-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s − 9-s + 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 1.43·31-s + 0.676·35-s − 0.986·37-s + 0.937·41-s − 1.21·43-s − 0.447·45-s + 0.583·47-s + 9/7·49-s + 0.824·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s − 1.51·63-s − 0.248·65-s − 0.977·67-s + 0.702·73-s + 1.82·77-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(67280\)    =    \(2^{4} \cdot 5 \cdot 29^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{67280} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 67280,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.999211191$
$L(\frac12)$  $\approx$  $2.999211191$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;29\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
29 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.32541099956365, −13.73461532301270, −13.46233991929693, −12.52652745638125, −11.98126680413034, −11.70613035411498, −11.27179641291055, −10.72070615823173, −10.21032581686732, −9.451110605942933, −9.004932102235330, −8.695636370409529, −7.968755441030361, −7.564327895602543, −6.927590612778440, −6.317146872662646, −5.662599663187747, −5.225012002253469, −4.778074072525518, −3.960208353886743, −3.514909565172452, −2.557314748687787, −1.979901771653407, −1.484352452114936, −0.5761170239314646, 0.5761170239314646, 1.484352452114936, 1.979901771653407, 2.557314748687787, 3.514909565172452, 3.960208353886743, 4.778074072525518, 5.225012002253469, 5.662599663187747, 6.317146872662646, 6.927590612778440, 7.564327895602543, 7.968755441030361, 8.695636370409529, 9.004932102235330, 9.451110605942933, 10.21032581686732, 10.72070615823173, 11.27179641291055, 11.70613035411498, 11.98126680413034, 12.52652745638125, 13.46233991929693, 13.73461532301270, 14.32541099956365

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