Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{4} \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  + 5-s + 7-s − 3·9-s − 6·11-s − 6·13-s − 4·17-s − 14·19-s − 23-s + 5·25-s − 2·29-s + 10·31-s + 35-s − 12·37-s + 8·41-s − 10·43-s − 3·45-s + 8·47-s + 4·53-s − 6·55-s − 7·61-s − 3·63-s − 6·65-s − 12·67-s − 30·71-s − 4·73-s − 6·77-s + 79-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 9-s − 1.80·11-s − 1.66·13-s − 0.970·17-s − 3.21·19-s − 0.208·23-s + 25-s − 0.371·29-s + 1.79·31-s + 0.169·35-s − 1.97·37-s + 1.24·41-s − 1.52·43-s − 0.447·45-s + 1.16·47-s + 0.549·53-s − 0.809·55-s − 0.896·61-s − 0.377·63-s − 0.744·65-s − 1.46·67-s − 3.56·71-s − 0.468·73-s − 0.683·77-s + 0.112·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.2313659624\)
\(L(\frac12)\)  \(\approx\)  \(0.2313659624\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−10.41334741536988331336256046221, −9.975751279554894950698185171397, −9.117398423675225066619430229700, −8.796483284033356513921534811987, −8.709501614789002665773507504302, −8.019502367706033541894703541010, −7.890511484569878029122806067196, −7.10000505504824821251188717567, −6.94412296329149120012633734727, −6.10039384850856176607967336767, −6.08688729004508515663104924858, −5.38471656937120029026824900334, −4.79050826078438600298430538030, −4.67022628736874351729717458984, −4.20506971988925797770657569798, −3.11584046593279726622395732545, −2.68656046954670565157106998634, −2.27866574135177906016483310826, −1.89066272747593138913215682996, −0.19802837887515089557502944622, 0.19802837887515089557502944622, 1.89066272747593138913215682996, 2.27866574135177906016483310826, 2.68656046954670565157106998634, 3.11584046593279726622395732545, 4.20506971988925797770657569798, 4.67022628736874351729717458984, 4.79050826078438600298430538030, 5.38471656937120029026824900334, 6.08688729004508515663104924858, 6.10039384850856176607967336767, 6.94412296329149120012633734727, 7.10000505504824821251188717567, 7.890511484569878029122806067196, 8.019502367706033541894703541010, 8.709501614789002665773507504302, 8.796483284033356513921534811987, 9.117398423675225066619430229700, 9.975751279554894950698185171397, 10.41334741536988331336256046221

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