Properties

Degree 4
Conductor $ 2^{12} \cdot 7^{2} $
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  + 4·3-s + 4·7-s + 6·9-s + 4·19-s + 16·21-s − 2·25-s − 4·27-s − 12·29-s + 16·31-s + 4·37-s + 9·49-s − 12·53-s + 16·57-s + 12·59-s + 24·63-s − 8·75-s − 37·81-s − 12·83-s − 48·87-s + 64·93-s − 8·103-s − 28·109-s + 16·111-s − 36·113-s + 10·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 2.30·3-s + 1.51·7-s + 2·9-s + 0.917·19-s + 3.49·21-s − 2/5·25-s − 0.769·27-s − 2.22·29-s + 2.87·31-s + 0.657·37-s + 9/7·49-s − 1.64·53-s + 2.11·57-s + 1.56·59-s + 3.02·63-s − 0.923·75-s − 4.11·81-s − 1.31·83-s − 5.14·87-s + 6.63·93-s − 0.788·103-s − 2.68·109-s + 1.51·111-s − 3.38·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(200704\)    =    \(2^{12} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{448} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 200704,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $4.34323$
$L(\frac12)$  $\approx$  $4.34323$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−11.25706804491268867726777168842, −11.01287155568625878927977115204, −10.26292381518721065546968140955, −9.588953830211663260248387217318, −9.550059819250600182882265266760, −9.036128236832154519105890323137, −8.345910353654609925558497678554, −8.247623890924368214748670444184, −7.907070679167903386448456167174, −7.52496880825596917724798965123, −6.98708042166248761297055838228, −6.16478904905507191535804434872, −5.50244752816487965820653538501, −5.08387460991897900681181111709, −4.17067380789998926272961730796, −3.99748740106848241050424196853, −3.08433232669730184582890431382, −2.73617141151098625327292214363, −2.05462471309512005043083227882, −1.40351027887794056703929038807, 1.40351027887794056703929038807, 2.05462471309512005043083227882, 2.73617141151098625327292214363, 3.08433232669730184582890431382, 3.99748740106848241050424196853, 4.17067380789998926272961730796, 5.08387460991897900681181111709, 5.50244752816487965820653538501, 6.16478904905507191535804434872, 6.98708042166248761297055838228, 7.52496880825596917724798965123, 7.907070679167903386448456167174, 8.247623890924368214748670444184, 8.345910353654609925558497678554, 9.036128236832154519105890323137, 9.550059819250600182882265266760, 9.588953830211663260248387217318, 10.26292381518721065546968140955, 11.01287155568625878927977115204, 11.25706804491268867726777168842

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