Properties

Degree 4
Conductor $ 2^{2} \cdot 11^{2} \cdot 19^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·5-s − 4·8-s + 6·9-s + 8·10-s + 2·11-s − 12·13-s + 5·16-s − 12·18-s − 6·19-s − 12·20-s − 4·22-s − 8·23-s + 2·25-s + 24·26-s + 12·29-s − 6·32-s + 18·36-s + 12·38-s + 16·40-s + 6·44-s − 24·45-s + 16·46-s − 16·47-s + 4·49-s − 4·50-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.78·5-s − 1.41·8-s + 2·9-s + 2.52·10-s + 0.603·11-s − 3.32·13-s + 5/4·16-s − 2.82·18-s − 1.37·19-s − 2.68·20-s − 0.852·22-s − 1.66·23-s + 2/5·25-s + 4.70·26-s + 2.22·29-s − 1.06·32-s + 3·36-s + 1.94·38-s + 2.52·40-s + 0.904·44-s − 3.57·45-s + 2.35·46-s − 2.33·47-s + 4/7·49-s − 0.565·50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(174724\)    =    \(2^{2} \cdot 11^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{174724} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 174724,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2976551487$
$L(\frac12)$  $\approx$  $0.2976551487$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + T )^{2} \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
19$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 112 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 52 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 126 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 166 T^{2} + p^{2} T^{4} \)
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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

−9.401015651062948175712097107372, −8.404698546837973896382080257851, −8.254241647099399101113112800091, −7.71388039206146437273757543984, −7.44762265446535733044525620099, −6.95010365868690331727234701514, −6.74088728555677034406075891110, −6.02811862778622137859657311885, −4.70625924775662572057444076338, −4.64997904029601505456448773943, −4.10836519063235601385128301951, −3.34818067703718862826595783894, −2.32771420129432717051475941594, −1.84693101155797292146430206660, −0.43010836589856531599598196145, 0.43010836589856531599598196145, 1.84693101155797292146430206660, 2.32771420129432717051475941594, 3.34818067703718862826595783894, 4.10836519063235601385128301951, 4.64997904029601505456448773943, 4.70625924775662572057444076338, 6.02811862778622137859657311885, 6.74088728555677034406075891110, 6.95010365868690331727234701514, 7.44762265446535733044525620099, 7.71388039206146437273757543984, 8.254241647099399101113112800091, 8.404698546837973896382080257851, 9.401015651062948175712097107372

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