Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 11^{3} $
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  + 2·2-s − 4-s − 8·8-s − 3·9-s − 11-s − 7·16-s + 12·17-s − 6·18-s − 2·22-s + 25-s + 12·29-s − 16·31-s + 14·32-s + 24·34-s + 3·36-s − 4·37-s + 4·41-s + 44-s − 14·49-s + 2·50-s + 24·58-s − 32·62-s + 35·64-s − 32·67-s − 12·68-s + 24·72-s − 8·74-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 2.82·8-s − 9-s − 0.301·11-s − 7/4·16-s + 2.91·17-s − 1.41·18-s − 0.426·22-s + 1/5·25-s + 2.22·29-s − 2.87·31-s + 2.47·32-s + 4.11·34-s + 1/2·36-s − 0.657·37-s + 0.624·41-s + 0.150·44-s − 2·49-s + 0.282·50-s + 3.15·58-s − 4.06·62-s + 35/8·64-s − 3.90·67-s − 1.45·68-s + 2.82·72-s − 0.929·74-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 299475 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 299475 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(299475\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{299475} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 299475,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_2$ \( 1 + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−8.551228798296751854049846045093, −8.248642768475844318802620430882, −7.65244290168268394743634881542, −7.25611852136541238398716522907, −6.16821463973303244938343944630, −6.08693701335829428157980562583, −5.54640979880007283696239996782, −5.09766059609940450216711100862, −4.87806253960044967314702320131, −4.10497602187160877381582357369, −3.40349928139289108990558562724, −3.25423179226253980082861279896, −2.73199597836315214499399961863, −1.27797148118613320465803316120, 0, 1.27797148118613320465803316120, 2.73199597836315214499399961863, 3.25423179226253980082861279896, 3.40349928139289108990558562724, 4.10497602187160877381582357369, 4.87806253960044967314702320131, 5.09766059609940450216711100862, 5.54640979880007283696239996782, 6.08693701335829428157980562583, 6.16821463973303244938343944630, 7.25611852136541238398716522907, 7.65244290168268394743634881542, 8.248642768475844318802620430882, 8.551228798296751854049846045093

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