Properties

Degree 8
Conductor $ 3^{4} \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  − 4·3-s − 2·4-s + 4·7-s + 6·9-s + 8·12-s − 16·13-s − 5·16-s − 16·21-s + 2·25-s + 4·27-s − 8·28-s − 12·36-s + 64·39-s + 20·48-s − 2·49-s + 32·52-s + 24·63-s + 20·64-s + 32·73-s − 8·75-s + 32·79-s − 37·81-s + 32·84-s − 64·91-s + 32·97-s − 4·100-s − 40·103-s + ⋯
L(s)  = 1  − 2.30·3-s − 4-s + 1.51·7-s + 2·9-s + 2.30·12-s − 4.43·13-s − 5/4·16-s − 3.49·21-s + 2/5·25-s + 0.769·27-s − 1.51·28-s − 2·36-s + 10.2·39-s + 2.88·48-s − 2/7·49-s + 4.43·52-s + 3.02·63-s + 5/2·64-s + 3.74·73-s − 0.923·75-s + 3.60·79-s − 4.11·81-s + 3.49·84-s − 6.70·91-s + 3.24·97-s − 2/5·100-s − 3.94·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{4} \cdot 5^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $0.166804$
$L(\frac12)$  $\approx$  $0.166804$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
89$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.19070171703711611628078833786, −9.813898458048051618481733890212, −9.690602564704510043371776154013, −9.480220212427536660710977679283, −9.254308968911655758612174884459, −8.787811672601936250810851613259, −8.336372474437866384699699670877, −8.187704968624703758502995805225, −7.64916586128995768211890223380, −7.61366997621558420853803955185, −7.16447954574929738174630481549, −6.68691565206057378813171168210, −6.60881536426890603723315013468, −6.40265755981170127584780735458, −5.57778899238268233473338911925, −5.22817115143474365843978716757, −5.11886651210099380061908742700, −4.92041555918160653379701026710, −4.84035481314464242271184792663, −4.38625364227784555644031732241, −4.01762483810657144674943936476, −3.01381048990394622851526182839, −2.34515833163741102245773879332, −2.13235648433281233133195991540, −0.47956781225255046972740175522, 0.47956781225255046972740175522, 2.13235648433281233133195991540, 2.34515833163741102245773879332, 3.01381048990394622851526182839, 4.01762483810657144674943936476, 4.38625364227784555644031732241, 4.84035481314464242271184792663, 4.92041555918160653379701026710, 5.11886651210099380061908742700, 5.22817115143474365843978716757, 5.57778899238268233473338911925, 6.40265755981170127584780735458, 6.60881536426890603723315013468, 6.68691565206057378813171168210, 7.16447954574929738174630481549, 7.61366997621558420853803955185, 7.64916586128995768211890223380, 8.187704968624703758502995805225, 8.336372474437866384699699670877, 8.787811672601936250810851613259, 9.254308968911655758612174884459, 9.480220212427536660710977679283, 9.690602564704510043371776154013, 9.813898458048051618481733890212, 10.19070171703711611628078833786

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