Properties

Degree 4
Conductor $ 2^{12} \cdot 3^{2} \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  − 2·3-s − 5-s + 3·9-s + 11-s + 5·13-s + 2·15-s + 8·17-s − 5·19-s + 8·23-s + 5·25-s − 4·27-s − 3·29-s + 2·31-s − 2·33-s − 3·37-s − 10·39-s − 6·41-s + 7·43-s − 3·45-s + 12·47-s − 16·51-s − 11·53-s − 55-s + 10·57-s + 5·59-s + 20·61-s − 5·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 9-s + 0.301·11-s + 1.38·13-s + 0.516·15-s + 1.94·17-s − 1.14·19-s + 1.66·23-s + 25-s − 0.769·27-s − 0.557·29-s + 0.359·31-s − 0.348·33-s − 0.493·37-s − 1.60·39-s − 0.937·41-s + 1.06·43-s − 0.447·45-s + 1.75·47-s − 2.24·51-s − 1.51·53-s − 0.134·55-s + 1.32·57-s + 0.650·59-s + 2.56·61-s − 0.620·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(88510464\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{9408} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.811534747\)
\(L(\frac12)\)  \(\approx\)  \(2.811534747\)
\(L(\frac{3}{2})\)   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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 3 T + 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 11 T + 122 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 7 T + 132 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - T + 132 T^{2} - p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 7 T + 164 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 130 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 25 T + 336 T^{2} + 25 p T^{3} + p^{2} T^{4} \)
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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

−7.76119634281237893591342427928, −7.52163005416145884995657889863, −7.08523174645885749201228108695, −6.70969853403009036705803696575, −6.58454568030360544510996201111, −6.26546714427569278029189793645, −5.62288555779463556181298404206, −5.53246300652978250830600078117, −5.28523032267949041717496278463, −4.86138220182874694601741093827, −4.29097249145847133527920600111, −4.13032497675797827736617601388, −3.62631428653427476413652113246, −3.41851975336195283532751203654, −2.89785362068929057017137540454, −2.45649385784649586130640903739, −1.69292327475941469974604554495, −1.35309396468921934033987633481, −0.805992375472753294090837094381, −0.59221846577716553688416621360, 0.59221846577716553688416621360, 0.805992375472753294090837094381, 1.35309396468921934033987633481, 1.69292327475941469974604554495, 2.45649385784649586130640903739, 2.89785362068929057017137540454, 3.41851975336195283532751203654, 3.62631428653427476413652113246, 4.13032497675797827736617601388, 4.29097249145847133527920600111, 4.86138220182874694601741093827, 5.28523032267949041717496278463, 5.53246300652978250830600078117, 5.62288555779463556181298404206, 6.26546714427569278029189793645, 6.58454568030360544510996201111, 6.70969853403009036705803696575, 7.08523174645885749201228108695, 7.52163005416145884995657889863, 7.76119634281237893591342427928

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