Properties

Degree 4
Conductor $ 2^{2} \cdot 3 \cdot 5^{2} \cdot 17^{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  + 3-s − 4-s + 2·5-s + 2·7-s − 12-s + 2·15-s + 16-s − 2·17-s + 8·19-s − 2·20-s + 2·21-s + 2·23-s − 25-s − 4·27-s − 2·28-s + 4·35-s + 8·37-s + 48-s − 2·49-s − 2·51-s + 8·57-s − 4·59-s − 2·60-s − 64-s + 2·68-s + 2·69-s + 8·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.755·7-s − 0.288·12-s + 0.516·15-s + 1/4·16-s − 0.485·17-s + 1.83·19-s − 0.447·20-s + 0.436·21-s + 0.417·23-s − 1/5·25-s − 0.769·27-s − 0.377·28-s + 0.676·35-s + 1.31·37-s + 0.144·48-s − 2/7·49-s − 0.280·51-s + 1.05·57-s − 0.520·59-s − 0.258·60-s − 1/8·64-s + 0.242·68-s + 0.240·69-s + 0.936·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(86700\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{86700} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 86700,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.108638445$
$L(\frac12)$  $\approx$  $2.108638445$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T^{2} \)
3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
17$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.515787212071587608976987326879, −9.286786413609602707355343813121, −8.814094502953020134316548756674, −8.121545273000048326248326912994, −7.80845566987432003768565197912, −7.32211078113343980519630660530, −6.61631553829504663083294799259, −5.93148042739259581781700849751, −5.47610983754170823461728105342, −4.95535452098970716543893127712, −4.35869357077627461302818756248, −3.58702899448501549862735144331, −2.88264402882531541396188854345, −2.10707937231367077654694312747, −1.22256098708743177778818610952, 1.22256098708743177778818610952, 2.10707937231367077654694312747, 2.88264402882531541396188854345, 3.58702899448501549862735144331, 4.35869357077627461302818756248, 4.95535452098970716543893127712, 5.47610983754170823461728105342, 5.93148042739259581781700849751, 6.61631553829504663083294799259, 7.32211078113343980519630660530, 7.80845566987432003768565197912, 8.121545273000048326248326912994, 8.814094502953020134316548756674, 9.286786413609602707355343813121, 9.515787212071587608976987326879

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