Properties

Degree $4$
Conductor $112896$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s − 5·7-s − 6·11-s − 6·13-s + 2·15-s − 4·17-s − 5·19-s + 5·21-s − 4·23-s + 5·25-s + 27-s − 8·29-s + 7·31-s + 6·33-s + 10·35-s + 9·37-s + 6·39-s − 4·41-s + 2·43-s + 2·47-s + 18·49-s + 4·51-s − 8·53-s + 12·55-s + 5·57-s − 10·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 1.88·7-s − 1.80·11-s − 1.66·13-s + 0.516·15-s − 0.970·17-s − 1.14·19-s + 1.09·21-s − 0.834·23-s + 25-s + 0.192·27-s − 1.48·29-s + 1.25·31-s + 1.04·33-s + 1.69·35-s + 1.47·37-s + 0.960·39-s − 0.624·41-s + 0.304·43-s + 0.291·47-s + 18/7·49-s + 0.560·51-s − 1.09·53-s + 1.61·55-s + 0.662·57-s − 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(112896\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{336} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 112896,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 8 T + 11 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 15 T + 158 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.10213719428926642074185763120, −10.99162948705489146055789648163, −10.35288409349239335369167326936, −10.05888590317596137748320286070, −9.498878150527566426318360980753, −9.209743753786910685072572051939, −8.315107524878971731134870671499, −8.054776784730632977802299399974, −7.27314291199556228480060329109, −7.20183996063302235064507655421, −6.23092964734093611090223021824, −6.22416097539569206251055165862, −5.35698645204664978849906927289, −4.73881430067421397360924156410, −4.30339836276678329012698749824, −3.53054952890293898441225217924, −2.59065877541454862571978846217, −2.53071278296397469934255790874, 0, 0, 2.53071278296397469934255790874, 2.59065877541454862571978846217, 3.53054952890293898441225217924, 4.30339836276678329012698749824, 4.73881430067421397360924156410, 5.35698645204664978849906927289, 6.22416097539569206251055165862, 6.23092964734093611090223021824, 7.20183996063302235064507655421, 7.27314291199556228480060329109, 8.054776784730632977802299399974, 8.315107524878971731134870671499, 9.209743753786910685072572051939, 9.498878150527566426318360980753, 10.05888590317596137748320286070, 10.35288409349239335369167326936, 10.99162948705489146055789648163, 11.10213719428926642074185763120

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