Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{2} \cdot 37^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 4-s + 2·7-s + 6·9-s − 4·12-s + 16-s − 8·21-s − 10·25-s + 4·27-s + 2·28-s + 6·36-s + 2·37-s + 12·41-s − 24·47-s − 4·48-s + 3·49-s + 12·53-s + 12·63-s + 64-s − 8·67-s + 4·73-s + 40·75-s − 37·81-s − 12·83-s − 8·84-s − 10·100-s + 24·107-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/2·4-s + 0.755·7-s + 2·9-s − 1.15·12-s + 1/4·16-s − 1.74·21-s − 2·25-s + 0.769·27-s + 0.377·28-s + 36-s + 0.328·37-s + 1.87·41-s − 3.50·47-s − 0.577·48-s + 3/7·49-s + 1.64·53-s + 1.51·63-s + 1/8·64-s − 0.977·67-s + 0.468·73-s + 4.61·75-s − 4.11·81-s − 1.31·83-s − 0.872·84-s − 100-s + 2.32·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 268324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 268324 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(268324\)    =    \(2^{2} \cdot 7^{2} \cdot 37^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{268324} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 268324,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;37\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;37\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( ( 1 - T )^{2} \)
37$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.485152769343560672100851205130, −8.169998867413597699811066785828, −7.57571100088867902110310233811, −7.16851316689893371261782075681, −6.45499171287041897802749746589, −6.26080129812508362760024888239, −5.61058899029416894793643414289, −5.57928681742950427486583645839, −4.81446031852430885616934703988, −4.47617404964643867507446377668, −3.72847359898829464933007778606, −2.82267866440533335693556734961, −1.97153093819951005096748677540, −1.10321404612740190875240729674, 0, 1.10321404612740190875240729674, 1.97153093819951005096748677540, 2.82267866440533335693556734961, 3.72847359898829464933007778606, 4.47617404964643867507446377668, 4.81446031852430885616934703988, 5.57928681742950427486583645839, 5.61058899029416894793643414289, 6.26080129812508362760024888239, 6.45499171287041897802749746589, 7.16851316689893371261782075681, 7.57571100088867902110310233811, 8.169998867413597699811066785828, 8.485152769343560672100851205130

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