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 1

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  =  1
Selberg data  =  $(4,\ 86700,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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

−9.602450395332515902653177427899, −8.903226050669653133498406827043, −8.470972699616657219188598063610, −7.86248062458750721035094601342, −7.40099724198824325946742762396, −7.00468487896800689290681838843, −6.26051044940577238616952398244, −5.77860674375791231685655396763, −5.17227258685475493351488259492, −4.72567853542324858839362308362, −3.84279492885884900122908818342, −3.46561426917601934500160503796, −2.78287239495500909169658277300, −1.28286948279600013610537350332, 0, 1.28286948279600013610537350332, 2.78287239495500909169658277300, 3.46561426917601934500160503796, 3.84279492885884900122908818342, 4.72567853542324858839362308362, 5.17227258685475493351488259492, 5.77860674375791231685655396763, 6.26051044940577238616952398244, 7.00468487896800689290681838843, 7.40099724198824325946742762396, 7.86248062458750721035094601342, 8.470972699616657219188598063610, 8.903226050669653133498406827043, 9.602450395332515902653177427899

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