Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 13^{2} $
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  − 2·3-s − 3·4-s + 3·9-s + 6·12-s − 2·13-s + 5·16-s + 4·17-s + 25-s − 4·27-s − 4·29-s − 9·36-s + 4·39-s + 8·43-s − 10·48-s − 14·49-s − 8·51-s + 6·52-s − 20·53-s − 4·61-s − 3·64-s − 12·68-s − 2·75-s + 5·81-s + 8·87-s − 3·100-s + 12·101-s − 32·103-s + ⋯
L(s)  = 1  − 1.15·3-s − 3/2·4-s + 9-s + 1.73·12-s − 0.554·13-s + 5/4·16-s + 0.970·17-s + 1/5·25-s − 0.769·27-s − 0.742·29-s − 3/2·36-s + 0.640·39-s + 1.21·43-s − 1.44·48-s − 2·49-s − 1.12·51-s + 0.832·52-s − 2.74·53-s − 0.512·61-s − 3/8·64-s − 1.45·68-s − 0.230·75-s + 5/9·81-s + 0.857·87-s − 0.299·100-s + 1.19·101-s − 3.15·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(38025\)    =    \(3^{2} \cdot 5^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{38025} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 38025,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

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

−9.910992058599380201232598700641, −9.523451675812284265792380668213, −9.333388465635554502086321327335, −8.526905706632633062975018882770, −7.76910750368042878757814163764, −7.66488013441745380243230842523, −6.63078439380576833175908493602, −6.20481649249773216903893847569, −5.27152076797277019902163228084, −5.23920392624592057055772361749, −4.46342990915686607443282092868, −3.92265287197659703286168465477, −3.01746980794826050429743741143, −1.41836019676092930181617969749, 0, 1.41836019676092930181617969749, 3.01746980794826050429743741143, 3.92265287197659703286168465477, 4.46342990915686607443282092868, 5.23920392624592057055772361749, 5.27152076797277019902163228084, 6.20481649249773216903893847569, 6.63078439380576833175908493602, 7.66488013441745380243230842523, 7.76910750368042878757814163764, 8.526905706632633062975018882770, 9.333388465635554502086321327335, 9.523451675812284265792380668213, 9.910992058599380201232598700641

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