Properties

Degree 4
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  + 4-s + 4·5-s + 7-s − 2·9-s − 5·13-s + 16-s + 4·20-s − 6·23-s + 8·25-s + 28-s − 3·29-s + 4·35-s − 2·36-s − 8·45-s − 11·49-s − 5·52-s − 6·59-s − 2·63-s + 64-s − 20·65-s + 13·67-s + 6·71-s + 4·80-s − 5·81-s + 9·83-s − 5·91-s − 6·92-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.78·5-s + 0.377·7-s − 2/3·9-s − 1.38·13-s + 1/4·16-s + 0.894·20-s − 1.25·23-s + 8/5·25-s + 0.188·28-s − 0.557·29-s + 0.676·35-s − 1/3·36-s − 1.19·45-s − 1.57·49-s − 0.693·52-s − 0.781·59-s − 0.251·63-s + 1/8·64-s − 2.48·65-s + 1.58·67-s + 0.712·71-s + 0.447·80-s − 5/9·81-s + 0.987·83-s − 0.524·91-s − 0.625·92-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(16820\)    =    \(2^{2} \cdot 5 \cdot 29^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{16820} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 16820,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.507833519$
$L(\frac12)$  $\approx$  $1.507833519$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;29\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;29\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 3 T + p T^{2} ) \)
29$C_2$ \( 1 + 3 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 128 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
89$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.08661603835906627439652366747, −10.23858562194629731076232750078, −10.03971941445509995315194425412, −9.507275604298581857199210722511, −9.018079617612541473926938785192, −8.216386704595267551433915339737, −7.70696143831198272236901713458, −6.98461397487466204179882331116, −6.27991464987397779791610023531, −5.87788579500999534452013345660, −5.24096448944400160426705236449, −4.66638385389512071647400360849, −3.37106367684779932314518719379, −2.35746149861219306678210936929, −1.90854538214432964525538723189, 1.90854538214432964525538723189, 2.35746149861219306678210936929, 3.37106367684779932314518719379, 4.66638385389512071647400360849, 5.24096448944400160426705236449, 5.87788579500999534452013345660, 6.27991464987397779791610023531, 6.98461397487466204179882331116, 7.70696143831198272236901713458, 8.216386704595267551433915339737, 9.018079617612541473926938785192, 9.507275604298581857199210722511, 10.03971941445509995315194425412, 10.23858562194629731076232750078, 11.08661603835906627439652366747

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