Properties

Degree 4
Conductor $ 2^{5} \cdot 3^{2} \cdot 31^{2} $
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 + 4-s + 8-s − 3·9-s + 4·13-s + 16-s − 3·18-s + 16·23-s − 6·25-s + 4·26-s + 32-s − 3·36-s + 20·37-s + 16·46-s − 16·47-s − 14·49-s − 6·50-s + 4·52-s − 24·59-s − 12·61-s + 64-s + 16·71-s − 3·72-s + 20·73-s + 20·74-s + 9·81-s + 16·83-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s + 1.10·13-s + 1/4·16-s − 0.707·18-s + 3.33·23-s − 6/5·25-s + 0.784·26-s + 0.176·32-s − 1/2·36-s + 3.28·37-s + 2.35·46-s − 2.33·47-s − 2·49-s − 0.848·50-s + 0.554·52-s − 3.12·59-s − 1.53·61-s + 1/8·64-s + 1.89·71-s − 0.353·72-s + 2.34·73-s + 2.32·74-s + 81-s + 1.75·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276768 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276768 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(276768\)    =    \(2^{5} \cdot 3^{2} \cdot 31^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{276768} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 276768,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.838960258\)
\(L(\frac12)\)  \(\approx\)  \(2.838960258\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;31\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 - T \)
3$C_2$ \( 1 + p T^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.056498267458590117851162621111, −8.228150792954604791756150599674, −7.78432113554336725186102006286, −7.76857292337562945056886541715, −6.68841524677704446114152278891, −6.32858579420662158347624019611, −6.20957515971754775976198016729, −5.43138638838697497763428859213, −4.76892395639548888503622113417, −4.74203912193695697715499891246, −3.68575806630196919369820756911, −3.16188554299090604324575557253, −2.90212986037219285929538328410, −1.88722882510241810597714442566, −0.944147784971163780613450180079, 0.944147784971163780613450180079, 1.88722882510241810597714442566, 2.90212986037219285929538328410, 3.16188554299090604324575557253, 3.68575806630196919369820756911, 4.74203912193695697715499891246, 4.76892395639548888503622113417, 5.43138638838697497763428859213, 6.20957515971754775976198016729, 6.32858579420662158347624019611, 6.68841524677704446114152278891, 7.76857292337562945056886541715, 7.78432113554336725186102006286, 8.228150792954604791756150599674, 9.056498267458590117851162621111

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