Properties

Degree 4
Conductor $ 2^{7} \cdot 7^{2} \cdot 11^{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  + 4·5-s + 2·7-s − 6·9-s + 4·11-s − 16·19-s + 2·25-s + 8·35-s − 4·37-s + 8·43-s − 24·45-s + 3·49-s + 12·53-s + 16·55-s − 12·63-s + 8·77-s − 32·79-s + 27·81-s − 16·83-s − 12·89-s − 64·95-s − 12·97-s − 24·99-s + 24·107-s + 4·113-s + 5·121-s − 28·125-s + 127-s + ⋯
L(s)  = 1  + 1.78·5-s + 0.755·7-s − 2·9-s + 1.20·11-s − 3.67·19-s + 2/5·25-s + 1.35·35-s − 0.657·37-s + 1.21·43-s − 3.57·45-s + 3/7·49-s + 1.64·53-s + 2.15·55-s − 1.51·63-s + 0.911·77-s − 3.60·79-s + 3·81-s − 1.75·83-s − 1.27·89-s − 6.56·95-s − 1.21·97-s − 2.41·99-s + 2.32·107-s + 0.376·113-s + 5/11·121-s − 2.50·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(758912\)    =    \(2^{7} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{758912} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 758912,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−8.423245666401170068214022844193, −7.69587064441186150768078543731, −6.88310002274215763839569601337, −6.67949194679031572918581545842, −5.94974241082854154889853475745, −5.82613210142107824134972732669, −5.72315905059004231786170242312, −4.86884684581528043387154729760, −4.18465932715971087267449515138, −4.03151585237256402081332864489, −2.98780927444278559643957768594, −2.30074942792092931708066021417, −2.14556791802447766709825303506, −1.44245425010921288807179765849, 0, 1.44245425010921288807179765849, 2.14556791802447766709825303506, 2.30074942792092931708066021417, 2.98780927444278559643957768594, 4.03151585237256402081332864489, 4.18465932715971087267449515138, 4.86884684581528043387154729760, 5.72315905059004231786170242312, 5.82613210142107824134972732669, 5.94974241082854154889853475745, 6.67949194679031572918581545842, 6.88310002274215763839569601337, 7.69587064441186150768078543731, 8.423245666401170068214022844193

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