Properties

Degree 2
Conductor $ 2^{2} \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  + 2·3-s − 5-s + 2·7-s + 9-s + 2·13-s − 2·15-s + 6·17-s + 4·19-s + 4·21-s + 6·23-s + 25-s − 4·27-s + 4·31-s − 2·35-s − 2·37-s + 4·39-s − 6·41-s + 10·43-s − 45-s + 6·47-s − 3·49-s + 12·51-s − 6·53-s + 8·57-s + 12·59-s − 2·61-s + 2·63-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.917·19-s + 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 0.718·31-s − 0.338·35-s − 0.328·37-s + 0.640·39-s − 0.937·41-s + 1.52·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s + 1.68·51-s − 0.824·53-s + 1.05·57-s + 1.56·59-s − 0.256·61-s + 0.251·63-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16820\)    =    \(2^{2} \cdot 5 \cdot 29^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{16820} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 16820,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.195786377$
$L(\frac12)$  $\approx$  $4.195786377$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.77497486915745, −15.25582452520766, −14.67238707740285, −14.28788405784851, −13.89528445451996, −13.23152846458340, −12.68258445955798, −11.82508069842703, −11.60807922718766, −10.83394756213898, −10.23365445716969, −9.535100593390327, −8.942688621097727, −8.483197285337311, −7.847760534236091, −7.546829682891877, −6.851649162780267, −5.832822396766763, −5.283430344596769, −4.543011599389774, −3.717085824744381, −3.199029104729932, −2.628545365498039, −1.564333829812903, −0.8926823574174181, 0.8926823574174181, 1.564333829812903, 2.628545365498039, 3.199029104729932, 3.717085824744381, 4.543011599389774, 5.283430344596769, 5.832822396766763, 6.851649162780267, 7.546829682891877, 7.847760534236091, 8.483197285337311, 8.942688621097727, 9.535100593390327, 10.23365445716969, 10.83394756213898, 11.60807922718766, 11.82508069842703, 12.68258445955798, 13.23152846458340, 13.89528445451996, 14.28788405784851, 14.67238707740285, 15.25582452520766, 15.77497486915745

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