Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{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  − 2-s + 8-s − 8·13-s − 16-s + 6·17-s − 2·19-s + 5·25-s + 8·26-s + 12·29-s + 4·31-s − 6·34-s − 2·37-s + 2·38-s − 12·41-s + 16·43-s − 12·47-s − 5·50-s + 6·53-s − 12·58-s − 6·59-s − 8·61-s − 4·62-s + 64-s + 4·67-s − 2·73-s + 2·74-s − 8·79-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.353·8-s − 2.21·13-s − 1/4·16-s + 1.45·17-s − 0.458·19-s + 25-s + 1.56·26-s + 2.22·29-s + 0.718·31-s − 1.02·34-s − 0.328·37-s + 0.324·38-s − 1.87·41-s + 2.43·43-s − 1.75·47-s − 0.707·50-s + 0.824·53-s − 1.57·58-s − 0.781·59-s − 1.02·61-s − 0.508·62-s + 1/8·64-s + 0.488·67-s − 0.234·73-s + 0.232·74-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{882} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 777924,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.00395$
$L(\frac12)$  $\approx$  $1.00395$
$L(\frac{3}{2})$   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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−10.24050168938634619756520511423, −9.947962399167162574763516562225, −9.562839283642452873150273824419, −9.202208327398030096372454610694, −8.579070835181114491590579675972, −8.290591895518752341408238230342, −7.893845413274031277357094778620, −7.43214040408875706848220213907, −7.04484775800697037631545065049, −6.61190330864623345980003208225, −6.16705628437093657687964839083, −5.25382606869139825437544321754, −5.22717740194178482115636502177, −4.48973455666388034502967389192, −4.30707667531637213604764354311, −3.13792778408072637693602168647, −2.96914043439670374576015188211, −2.27167569372610041124362855296, −1.41601001243632920313198798025, −0.57480985419816511141810679089, 0.57480985419816511141810679089, 1.41601001243632920313198798025, 2.27167569372610041124362855296, 2.96914043439670374576015188211, 3.13792778408072637693602168647, 4.30707667531637213604764354311, 4.48973455666388034502967389192, 5.22717740194178482115636502177, 5.25382606869139825437544321754, 6.16705628437093657687964839083, 6.61190330864623345980003208225, 7.04484775800697037631545065049, 7.43214040408875706848220213907, 7.893845413274031277357094778620, 8.290591895518752341408238230342, 8.579070835181114491590579675972, 9.202208327398030096372454610694, 9.562839283642452873150273824419, 9.947962399167162574763516562225, 10.24050168938634619756520511423

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