Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 7^{3} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s − 2·5-s + 7-s + 9-s − 12-s + 2·15-s + 16-s + 12·17-s − 2·20-s − 21-s + 3·25-s − 27-s + 28-s − 2·35-s + 36-s − 20·37-s + 12·41-s − 8·43-s − 2·45-s − 48-s + 49-s − 12·51-s + 24·59-s + 2·60-s + 63-s + 64-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.288·12-s + 0.516·15-s + 1/4·16-s + 2.91·17-s − 0.447·20-s − 0.218·21-s + 3/5·25-s − 0.192·27-s + 0.188·28-s − 0.338·35-s + 1/6·36-s − 3.28·37-s + 1.87·41-s − 1.21·43-s − 0.298·45-s − 0.144·48-s + 1/7·49-s − 1.68·51-s + 3.12·59-s + 0.258·60-s + 0.125·63-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 926100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 926100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(926100\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 7^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{926100} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 926100,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.784008679$
$L(\frac12)$  $\approx$  $1.784008679$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 + T \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( 1 - T \)
good11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.065715785764846048787694632217, −7.67845508591579519153392227466, −7.19801386358340821477645347390, −7.06224867938800248389577305819, −6.45416331724144396581871011345, −5.76093596813527562032392850894, −5.44446745120546824484392539368, −5.20272031164140682979668587117, −4.51192569173011558819556455876, −3.88273199773070623460496754838, −3.39965389205971738905570150540, −3.13125557587754062819456448397, −2.13174907534489392599600946706, −1.41006155845401754086894809680, −0.70440313958741481965056319183, 0.70440313958741481965056319183, 1.41006155845401754086894809680, 2.13174907534489392599600946706, 3.13125557587754062819456448397, 3.39965389205971738905570150540, 3.88273199773070623460496754838, 4.51192569173011558819556455876, 5.20272031164140682979668587117, 5.44446745120546824484392539368, 5.76093596813527562032392850894, 6.45416331724144396581871011345, 7.06224867938800248389577305819, 7.19801386358340821477645347390, 7.67845508591579519153392227466, 8.065715785764846048787694632217

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