Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{2} \cdot 11^{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  − 4·3-s + 4-s + 6·9-s − 4·12-s + 16-s − 10·25-s + 4·27-s − 8·31-s + 6·36-s + 4·37-s − 24·47-s − 4·48-s + 49-s + 12·53-s − 12·59-s + 64-s − 8·67-s + 40·75-s − 37·81-s − 12·89-s + 32·93-s − 20·97-s − 10·100-s − 8·103-s + 4·108-s − 16·111-s + 12·113-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/2·4-s + 2·9-s − 1.15·12-s + 1/4·16-s − 2·25-s + 0.769·27-s − 1.43·31-s + 36-s + 0.657·37-s − 3.50·47-s − 0.577·48-s + 1/7·49-s + 1.64·53-s − 1.56·59-s + 1/8·64-s − 0.977·67-s + 4.61·75-s − 4.11·81-s − 1.27·89-s + 3.31·93-s − 2.03·97-s − 100-s − 0.788·103-s + 0.384·108-s − 1.51·111-s + 1.12·113-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(23716\)    =    \(2^{2} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{23716} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 23716,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;11\}$, 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$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−10.74827747141930613672865179341, −10.01782448311884311699973230891, −9.765547119459919407856461234632, −8.888226006934558931566258496787, −8.145906800879012290031802042193, −7.57571100088867902110310233811, −6.82397964191468499058519060303, −6.42455331216463290634831658204, −5.76321350617402115136418955935, −5.57928681742950427486583645839, −4.85990751466979898816213085515, −4.09052365620160024762211243831, −3.03039761753313177141617611476, −1.67120553980965395706477292535, 0, 1.67120553980965395706477292535, 3.03039761753313177141617611476, 4.09052365620160024762211243831, 4.85990751466979898816213085515, 5.57928681742950427486583645839, 5.76321350617402115136418955935, 6.42455331216463290634831658204, 6.82397964191468499058519060303, 7.57571100088867902110310233811, 8.145906800879012290031802042193, 8.888226006934558931566258496787, 9.765547119459919407856461234632, 10.01782448311884311699973230891, 10.74827747141930613672865179341

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