Properties

Degree 4
Conductor $ 2^{5} \cdot 7^{2} \cdot 11^{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 + 2·7-s − 8-s − 2·9-s − 2·14-s + 16-s + 2·18-s + 4·19-s − 10·25-s + 2·28-s − 32-s − 2·36-s + 4·37-s − 4·38-s + 16·43-s + 3·49-s + 10·50-s + 12·53-s − 2·56-s − 4·63-s + 64-s + 2·72-s − 4·74-s + 4·76-s + 16·79-s − 5·81-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.534·14-s + 1/4·16-s + 0.471·18-s + 0.917·19-s − 2·25-s + 0.377·28-s − 0.176·32-s − 1/3·36-s + 0.657·37-s − 0.648·38-s + 2.43·43-s + 3/7·49-s + 1.41·50-s + 1.64·53-s − 0.267·56-s − 0.503·63-s + 1/8·64-s + 0.235·72-s − 0.464·74-s + 0.458·76-s + 1.80·79-s − 5/9·81-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(189728\)    =    \(2^{5} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{189728} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 189728,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.183647893$
$L(\frac12)$  $\approx$  $1.183647893$
$L(\frac{3}{2})$   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$C_1$ \( 1 + T \)
7$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−9.139513650481559174344623349620, −8.527724601400848222788670647982, −8.305301817736640156017703661439, −7.57571100088867902110310233811, −7.52725309122624587441708234050, −6.87975810404488340043398375902, −6.10121749499441085588030360800, −5.57928681742950427486583645839, −5.51487510751237752431573865412, −4.42789561057506381074928638435, −4.05731801132236751489021266456, −3.20671223694547121384446442813, −2.49822445009788286784730652330, −1.86250960874307840989277735252, −0.795913931859384347135515195825, 0.795913931859384347135515195825, 1.86250960874307840989277735252, 2.49822445009788286784730652330, 3.20671223694547121384446442813, 4.05731801132236751489021266456, 4.42789561057506381074928638435, 5.51487510751237752431573865412, 5.57928681742950427486583645839, 6.10121749499441085588030360800, 6.87975810404488340043398375902, 7.52725309122624587441708234050, 7.57571100088867902110310233811, 8.305301817736640156017703661439, 8.527724601400848222788670647982, 9.139513650481559174344623349620

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