Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 7^{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  + 3-s + 4-s + 2·5-s + 7-s + 9-s + 12-s + 2·15-s + 16-s − 12·17-s + 2·20-s + 21-s + 3·25-s + 27-s + 28-s + 2·35-s + 36-s − 20·37-s − 12·41-s − 8·43-s + 2·45-s + 48-s + 49-s − 12·51-s − 24·59-s + 2·60-s + 63-s + 64-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.288·12-s + 0.516·15-s + 1/4·16-s − 2.91·17-s + 0.447·20-s + 0.218·21-s + 3/5·25-s + 0.192·27-s + 0.188·28-s + 0.338·35-s + 1/6·36-s − 3.28·37-s − 1.87·41-s − 1.21·43-s + 0.298·45-s + 0.144·48-s + 1/7·49-s − 1.68·51-s − 3.12·59-s + 0.258·60-s + 0.125·63-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 926100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 926100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(926100\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 7^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{926100} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 926100,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 - T \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( 1 - T \)
good11$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−7.988751310608497948309425656861, −7.48230039465939347417283743898, −6.88089320758720461322150711491, −6.63010526050831562050636982417, −6.43968643454495280602009146366, −5.75531801300200717416192667829, −4.97103303999261607614028493172, −4.92567559830316004423556337525, −4.33770231717830777250549003341, −3.45283173381757043345013171615, −3.22977772739188491587716942834, −2.33971670719236581491979893835, −1.84101162653270326557570577442, −1.69594451483764738863265937021, 0, 1.69594451483764738863265937021, 1.84101162653270326557570577442, 2.33971670719236581491979893835, 3.22977772739188491587716942834, 3.45283173381757043345013171615, 4.33770231717830777250549003341, 4.92567559830316004423556337525, 4.97103303999261607614028493172, 5.75531801300200717416192667829, 6.43968643454495280602009146366, 6.63010526050831562050636982417, 6.88089320758720461322150711491, 7.48230039465939347417283743898, 7.988751310608497948309425656861

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