Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{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  + 2·3-s + 4-s + 3·9-s + 2·12-s + 2·13-s + 16-s + 12·17-s + 25-s + 4·27-s − 12·29-s + 3·36-s + 4·39-s − 8·43-s + 2·48-s + 2·49-s + 24·51-s + 2·52-s − 12·53-s − 20·61-s + 64-s + 12·68-s + 2·75-s + 16·79-s + 5·81-s − 24·87-s + 100-s + 36·101-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s + 9-s + 0.577·12-s + 0.554·13-s + 1/4·16-s + 2.91·17-s + 1/5·25-s + 0.769·27-s − 2.22·29-s + 1/2·36-s + 0.640·39-s − 1.21·43-s + 0.288·48-s + 2/7·49-s + 3.36·51-s + 0.277·52-s − 1.64·53-s − 2.56·61-s + 1/8·64-s + 1.45·68-s + 0.230·75-s + 1.80·79-s + 5/9·81-s − 2.57·87-s + 1/10·100-s + 3.58·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(152100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{152100} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 152100,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.116182873\)
\(L(\frac12)\)  \(\approx\)  \(3.116182873\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;13\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.305587122869086271308905422277, −8.925260031303717303381313539625, −7.941231322257043910223245152113, −7.889742230800007366384340287619, −7.70424926324545431494851724983, −6.90106282892871459571015329925, −6.42217617652666865799421983967, −5.66012343286953819079657082523, −5.41509779315100847068653879171, −4.57317770756600442017159530038, −3.70681432856719735440542242113, −3.36585804145949552210873743484, −2.95923023255871163375029903372, −1.87988664142882943456143061246, −1.32801143307487185867201245142, 1.32801143307487185867201245142, 1.87988664142882943456143061246, 2.95923023255871163375029903372, 3.36585804145949552210873743484, 3.70681432856719735440542242113, 4.57317770756600442017159530038, 5.41509779315100847068653879171, 5.66012343286953819079657082523, 6.42217617652666865799421983967, 6.90106282892871459571015329925, 7.70424926324545431494851724983, 7.889742230800007366384340287619, 7.941231322257043910223245152113, 8.925260031303717303381313539625, 9.305587122869086271308905422277

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