Properties

Degree 4
Conductor $ 5^{2} \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  + 2·2-s − 4-s − 8·8-s − 6·9-s − 2·11-s − 7·16-s − 12·18-s − 4·22-s + 8·23-s + 25-s + 12·29-s + 14·32-s + 6·36-s − 4·37-s + 8·43-s + 2·44-s + 16·46-s − 7·49-s + 2·50-s − 4·53-s + 24·58-s + 35·64-s − 32·67-s + 16·71-s + 48·72-s − 8·74-s + 16·79-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 2.82·8-s − 2·9-s − 0.603·11-s − 7/4·16-s − 2.82·18-s − 0.852·22-s + 1.66·23-s + 1/5·25-s + 2.22·29-s + 2.47·32-s + 36-s − 0.657·37-s + 1.21·43-s + 0.301·44-s + 2.35·46-s − 49-s + 0.282·50-s − 0.549·53-s + 3.15·58-s + 35/8·64-s − 3.90·67-s + 1.89·71-s + 5.65·72-s − 0.929·74-s + 1.80·79-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(148225\)    =    \(5^{2} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{148225} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 148225,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.338464506$
$L(\frac12)$  $\approx$  $1.338464506$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + p T^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−9.365267708430525132762162914792, −8.781700662013046394929624444870, −8.248642768475844318802620430882, −8.240174245715852192939095672164, −7.24720948442492513050045074918, −6.40684005531684016854972661238, −6.16821463973303244938343944630, −5.57003540112200356703114172743, −5.09766059609940450216711100862, −4.82709630451403100091257393731, −4.26300768017318841213602111936, −3.25423179226253980082861279896, −3.10898016437346903231946450096, −2.59341539336063966352674092067, −0.61786367914774168667137108260, 0.61786367914774168667137108260, 2.59341539336063966352674092067, 3.10898016437346903231946450096, 3.25423179226253980082861279896, 4.26300768017318841213602111936, 4.82709630451403100091257393731, 5.09766059609940450216711100862, 5.57003540112200356703114172743, 6.16821463973303244938343944630, 6.40684005531684016854972661238, 7.24720948442492513050045074918, 8.240174245715852192939095672164, 8.248642768475844318802620430882, 8.781700662013046394929624444870, 9.365267708430525132762162914792

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