Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{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  − 2·3-s + 4-s + 2·7-s + 9-s − 2·12-s − 8·13-s + 16-s + 4·19-s − 4·21-s − 10·25-s + 4·27-s + 2·28-s − 8·31-s + 36-s + 4·37-s + 16·39-s + 16·43-s − 2·48-s + 3·49-s − 8·52-s − 8·57-s + 16·61-s + 2·63-s + 64-s − 8·67-s + 4·73-s + 20·75-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s − 0.577·12-s − 2.21·13-s + 1/4·16-s + 0.917·19-s − 0.872·21-s − 2·25-s + 0.769·27-s + 0.377·28-s − 1.43·31-s + 1/6·36-s + 0.657·37-s + 2.56·39-s + 2.43·43-s − 0.288·48-s + 3/7·49-s − 1.10·52-s − 1.05·57-s + 2.04·61-s + 0.251·63-s + 1/8·64-s − 0.977·67-s + 0.468·73-s + 2.30·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1764} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1764,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5054223186$
$L(\frac12)$  $\approx$  $0.5054223186$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−13.45852272915611541668421078683, −12.37171571172372853840948077604, −12.30526099005271094386573698884, −11.52629384186915375545261187858, −11.23136141438460806904855801814, −10.47027853370113893616936981615, −9.765547119459919407856461234632, −9.207333736191082013174831123730, −7.80365099617056766338064266882, −7.57571100088867902110310233811, −6.71310127772875365703449938682, −5.57928681742950427486583645839, −5.31921059379060528260176389250, −4.19102102405251164216473944262, −2.39101862347891857041372548490, 2.39101862347891857041372548490, 4.19102102405251164216473944262, 5.31921059379060528260176389250, 5.57928681742950427486583645839, 6.71310127772875365703449938682, 7.57571100088867902110310233811, 7.80365099617056766338064266882, 9.207333736191082013174831123730, 9.765547119459919407856461234632, 10.47027853370113893616936981615, 11.23136141438460806904855801814, 11.52629384186915375545261187858, 12.30526099005271094386573698884, 12.37171571172372853840948077604, 13.45852272915611541668421078683

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