Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{2} \cdot 7^{2} $
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 − 3·5-s − 5·7-s − 9·11-s + 3·15-s + 6·17-s + 2·19-s + 5·21-s − 12·23-s + 25-s + 27-s − 18·29-s + 31-s + 9·33-s + 15·35-s + 2·37-s + 18·49-s − 6·51-s − 9·53-s + 27·55-s − 2·57-s − 3·59-s − 12·61-s + 12·69-s + 12·73-s − 75-s + 45·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 1.88·7-s − 2.71·11-s + 0.774·15-s + 1.45·17-s + 0.458·19-s + 1.09·21-s − 2.50·23-s + 1/5·25-s + 0.192·27-s − 3.34·29-s + 0.179·31-s + 1.56·33-s + 2.53·35-s + 0.328·37-s + 18/7·49-s − 0.840·51-s − 1.23·53-s + 3.64·55-s − 0.264·57-s − 0.390·59-s − 1.53·61-s + 1.44·69-s + 1.40·73-s − 0.115·75-s + 5.12·77-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

\( d \)  =  \(4\)
\( N \)  =  \(112896\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{336} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 112896,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \)
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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

−11.18605671006109212954167084842, −10.95998550160901359027347475738, −10.34631137490228176771122800968, −10.04201671945893653801052312456, −9.545257984635324886593211677281, −9.259414738087848330733795032145, −8.079031798596557269617026778079, −7.934743136864111534373675017690, −7.62653990032028339360052445859, −7.24645603807802844624133882838, −6.28642125847517367664409703856, −5.85563143382967819386074800199, −5.53124706110679993099122855047, −4.94594415883703585734042969171, −3.87684834127316412962556989560, −3.64276965826824894169972705290, −3.02232996053267043969500318282, −2.20627190757909835975655245146, 0, 0, 2.20627190757909835975655245146, 3.02232996053267043969500318282, 3.64276965826824894169972705290, 3.87684834127316412962556989560, 4.94594415883703585734042969171, 5.53124706110679993099122855047, 5.85563143382967819386074800199, 6.28642125847517367664409703856, 7.24645603807802844624133882838, 7.62653990032028339360052445859, 7.934743136864111534373675017690, 8.079031798596557269617026778079, 9.259414738087848330733795032145, 9.545257984635324886593211677281, 10.04201671945893653801052312456, 10.34631137490228176771122800968, 10.95998550160901359027347475738, 11.18605671006109212954167084842

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