Properties

Degree 4
Conductor $ 2^{4} \cdot 37^{2} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s + 9-s − 2·17-s + 2·19-s − 2·23-s − 2·29-s + 2·47-s + 49-s + 2·53-s − 2·63-s + 2·71-s − 2·79-s − 2·83-s − 2·89-s − 2·109-s + 2·113-s + 4·119-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + ⋯
L(s)  = 1  − 2·7-s + 9-s − 2·17-s + 2·19-s − 2·23-s − 2·29-s + 2·47-s + 49-s + 2·53-s − 2·63-s + 2·71-s − 2·79-s − 2·83-s − 2·89-s − 2·109-s + 2·113-s + 4·119-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(21904\)    =    \(2^{4} \cdot 37^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{148} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 21904,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.3684437083$
$L(\frac12)$  $\approx$  $0.3684437083$
$L(1)$   not available
$L(1)$   not available

Euler product

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

−13.49148832149289409247800683737, −13.10816457337357686041792780832, −12.43625305431146567598223854948, −12.38211825179779823233357558045, −11.37649815088232660667564705338, −11.23585659445774750887200017992, −10.19320403310866793307351058503, −9.990264392653704431458007765550, −9.522405592957802838890433163166, −9.141281680914141676207393897133, −8.444784212317855955232376186194, −7.56795461850848674019017603429, −6.94968605451004120504551546434, −6.89050957330215446206173448352, −5.77021853794458276231113710756, −5.69019913107289534205610998386, −4.27714347876596393041059830264, −3.95290976747288843243538632860, −3.09928043927083506759952653755, −2.08628502080588857118603629212, 2.08628502080588857118603629212, 3.09928043927083506759952653755, 3.95290976747288843243538632860, 4.27714347876596393041059830264, 5.69019913107289534205610998386, 5.77021853794458276231113710756, 6.89050957330215446206173448352, 6.94968605451004120504551546434, 7.56795461850848674019017603429, 8.444784212317855955232376186194, 9.141281680914141676207393897133, 9.522405592957802838890433163166, 9.990264392653704431458007765550, 10.19320403310866793307351058503, 11.23585659445774750887200017992, 11.37649815088232660667564705338, 12.38211825179779823233357558045, 12.43625305431146567598223854948, 13.10816457337357686041792780832, 13.49148832149289409247800683737

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