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  =  $(4,\ 777924,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.68244$
$L(\frac12)$  $\approx$  $1.68244$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
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

−10.39247262359629443990942940583, −10.03458217196504737449000820073, −9.174995102540854900453790019561, −9.077121502901718740174244911167, −8.657303558006121919107339150908, −8.584248611012648524612602635586, −7.82916649232246112174780567500, −7.50832315445137166254741236384, −6.86895803606696537734860737541, −6.63343478235961428792921448827, −5.91763966675191839057885172732, −5.83842875049177632595080084384, −5.01874547052489253303943963197, −4.47193653024812030918892525950, −3.96201716102204789433231923865, −3.67099487377760487735754976729, −2.58645387986878904012478155109, −2.44197013065682108198827134917, −1.13998151026283401648955805326, −0.905061614027266334376927730819, 0.905061614027266334376927730819, 1.13998151026283401648955805326, 2.44197013065682108198827134917, 2.58645387986878904012478155109, 3.67099487377760487735754976729, 3.96201716102204789433231923865, 4.47193653024812030918892525950, 5.01874547052489253303943963197, 5.83842875049177632595080084384, 5.91763966675191839057885172732, 6.63343478235961428792921448827, 6.86895803606696537734860737541, 7.50832315445137166254741236384, 7.82916649232246112174780567500, 8.584248611012648524612602635586, 8.657303558006121919107339150908, 9.077121502901718740174244911167, 9.174995102540854900453790019561, 10.03458217196504737449000820073, 10.39247262359629443990942940583

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