Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{2} \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s + 6·11-s + 3·13-s + 2·25-s + 4·27-s − 12·33-s − 8·37-s − 6·39-s + 6·47-s + 2·49-s − 18·59-s − 20·61-s − 6·71-s + 16·73-s − 4·75-s − 11·81-s − 6·83-s + 4·97-s + 6·99-s − 12·107-s + 4·109-s + 16·111-s + 3·117-s + 14·121-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 1.80·11-s + 0.832·13-s + 2/5·25-s + 0.769·27-s − 2.08·33-s − 1.31·37-s − 0.960·39-s + 0.875·47-s + 2/7·49-s − 2.34·59-s − 2.56·61-s − 0.712·71-s + 1.87·73-s − 0.461·75-s − 1.22·81-s − 0.658·83-s + 0.406·97-s + 0.603·99-s − 1.16·107-s + 0.383·109-s + 1.51·111-s + 0.277·117-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(7488\)    =    \(2^{6} \cdot 3^{2} \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7488} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 7488,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.7067454387$
$L(\frac12)$  $\approx$  $0.7067454387$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;13\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;13\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−11.87155116813077068640664381906, −11.22223741775793504704253476270, −10.81731167510890640934599763781, −10.37515220498659616000734567466, −9.408432747001542418489178532692, −9.028704331668933074461162446989, −8.456296581918437833902757190517, −7.50956221473763168283760347313, −6.76577819406238587413747469202, −6.27901981478651695572674737545, −5.80690615762205488723594748732, −4.88918895983617812429598939925, −4.14856945798263111009867585329, −3.24989607732100239535744615059, −1.41770899276333058782442932324, 1.41770899276333058782442932324, 3.24989607732100239535744615059, 4.14856945798263111009867585329, 4.88918895983617812429598939925, 5.80690615762205488723594748732, 6.27901981478651695572674737545, 6.76577819406238587413747469202, 7.50956221473763168283760347313, 8.456296581918437833902757190517, 9.028704331668933074461162446989, 9.408432747001542418489178532692, 10.37515220498659616000734567466, 10.81731167510890640934599763781, 11.22223741775793504704253476270, 11.87155116813077068640664381906

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