Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{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  − 4·3-s + 4-s + 6·9-s − 4·12-s − 4·13-s + 16-s + 12·17-s − 10·25-s + 4·27-s − 12·29-s + 6·36-s + 16·39-s + 16·43-s − 4·48-s + 49-s − 48·51-s − 4·52-s + 12·53-s + 16·61-s + 64-s + 12·68-s + 40·75-s + 16·79-s − 37·81-s + 48·87-s − 10·100-s − 8·103-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/2·4-s + 2·9-s − 1.15·12-s − 1.10·13-s + 1/4·16-s + 2.91·17-s − 2·25-s + 0.769·27-s − 2.22·29-s + 36-s + 2.56·39-s + 2.43·43-s − 0.577·48-s + 1/7·49-s − 6.72·51-s − 0.554·52-s + 1.64·53-s + 2.04·61-s + 1/8·64-s + 1.45·68-s + 4.61·75-s + 1.80·79-s − 4.11·81-s + 5.14·87-s − 100-s − 0.788·103-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 33124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 33124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(33124\)    =    \(2^{2} \cdot 7^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{33124} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 33124,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5443988515$
$L(\frac12)$  $\approx$  $0.5443988515$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;13\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;13\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−10.49592006325651541786194815082, −10.05203493927597549730917438719, −9.765547119459919407856461234632, −9.048963436684934837350190246147, −7.985543387818251367130758863278, −7.57571100088867902110310233811, −7.21314366054207864456422661616, −6.39830303827760347609498636850, −5.67659105640301163348975864880, −5.57928681742950427486583645839, −5.32009626342602375518070026112, −4.20093704083038633652643494480, −3.42971528113344384838316462715, −2.23652265123208151083734264406, −0.78150816281722635549710851356, 0.78150816281722635549710851356, 2.23652265123208151083734264406, 3.42971528113344384838316462715, 4.20093704083038633652643494480, 5.32009626342602375518070026112, 5.57928681742950427486583645839, 5.67659105640301163348975864880, 6.39830303827760347609498636850, 7.21314366054207864456422661616, 7.57571100088867902110310233811, 7.985543387818251367130758863278, 9.048963436684934837350190246147, 9.765547119459919407856461234632, 10.05203493927597549730917438719, 10.49592006325651541786194815082

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