Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{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-s − 4-s + 2·5-s + 3·8-s + 9-s − 2·10-s − 4·13-s − 16-s + 4·17-s − 18-s − 2·20-s + 3·25-s + 4·26-s − 4·29-s − 5·32-s − 4·34-s − 36-s − 20·37-s + 6·40-s + 20·41-s + 2·45-s − 14·49-s − 3·50-s + 4·52-s − 20·53-s + 4·58-s − 4·61-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.10·13-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.447·20-s + 3/5·25-s + 0.784·26-s − 0.742·29-s − 0.883·32-s − 0.685·34-s − 1/6·36-s − 3.28·37-s + 0.948·40-s + 3.12·41-s + 0.298·45-s − 2·49-s − 0.424·50-s + 0.554·52-s − 2.74·53-s + 0.525·58-s − 0.512·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 3600,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5589254281$
$L(\frac12)$  $\approx$  $0.5589254281$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−12.64617876135106218873747419153, −12.26787268519799195488690762647, −11.16070118695890574879383709815, −10.67892245123374144028858824682, −10.00510556501914029619044642425, −9.523451675812284265792380668213, −9.265565204604989643425731491439, −8.330811446905088069540074876671, −7.66488013441745380243230842523, −7.12360478295630625617848815878, −6.13587545605028508104380795249, −5.23920392624592057055772361749, −4.68831052899409581492283307982, −3.39274372810076015084631955767, −1.82346513010869405208007428430, 1.82346513010869405208007428430, 3.39274372810076015084631955767, 4.68831052899409581492283307982, 5.23920392624592057055772361749, 6.13587545605028508104380795249, 7.12360478295630625617848815878, 7.66488013441745380243230842523, 8.330811446905088069540074876671, 9.265565204604989643425731491439, 9.523451675812284265792380668213, 10.00510556501914029619044642425, 10.67892245123374144028858824682, 11.16070118695890574879383709815, 12.26787268519799195488690762647, 12.64617876135106218873747419153

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