Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{4} \cdot 7^{4} $
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-s + 2·9-s + 16-s + 4·19-s + 12·29-s + 8·31-s − 2·36-s − 12·41-s − 12·59-s − 16·61-s − 64-s − 4·76-s − 16·79-s − 5·81-s − 12·89-s − 4·109-s − 12·116-s − 22·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1/2·4-s + 2/3·9-s + 1/4·16-s + 0.917·19-s + 2.22·29-s + 1.43·31-s − 1/3·36-s − 1.87·41-s − 1.56·59-s − 2.04·61-s − 1/8·64-s − 0.458·76-s − 1.80·79-s − 5/9·81-s − 1.27·89-s − 0.383·109-s − 1.11·116-s − 2·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(6002500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{2450} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 6002500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.018939629$
$L(\frac12)$  $\approx$  $2.018939629$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−9.291580408060141983979399186092, −8.686799213988998412180612357826, −8.372317926425118838681390711910, −7.974685124744191518425915416230, −7.77997128109805632868382916423, −7.07256928059767764542463318469, −6.94757629042622922564060045741, −6.30169340373003154799973259917, −6.26241248410960786189616124972, −5.41015405814042970595648062909, −5.29080240231530046836053673283, −4.52680662177885794476002978576, −4.52338079867262475696931665972, −4.08804702133336435501210645596, −3.17540836021014981116902664595, −3.10579981581082720866300495059, −2.60263928118040302973129514711, −1.54539346394132368876531348887, −1.39715208919112226567774463728, −0.51355078800606068609580219707, 0.51355078800606068609580219707, 1.39715208919112226567774463728, 1.54539346394132368876531348887, 2.60263928118040302973129514711, 3.10579981581082720866300495059, 3.17540836021014981116902664595, 4.08804702133336435501210645596, 4.52338079867262475696931665972, 4.52680662177885794476002978576, 5.29080240231530046836053673283, 5.41015405814042970595648062909, 6.26241248410960786189616124972, 6.30169340373003154799973259917, 6.94757629042622922564060045741, 7.07256928059767764542463318469, 7.77997128109805632868382916423, 7.974685124744191518425915416230, 8.372317926425118838681390711910, 8.686799213988998412180612357826, 9.291580408060141983979399186092

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