Properties

Degree $4$
Conductor $56448$
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 + 3·9-s + 8·19-s − 6·25-s + 4·27-s + 12·29-s − 16·31-s + 12·37-s − 7·49-s − 4·53-s + 16·57-s − 8·59-s − 12·75-s + 5·81-s + 8·83-s + 24·87-s − 32·93-s − 32·103-s − 4·109-s + 24·111-s + 36·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s − 14·147-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 1.83·19-s − 6/5·25-s + 0.769·27-s + 2.22·29-s − 2.87·31-s + 1.97·37-s − 49-s − 0.549·53-s + 2.11·57-s − 1.04·59-s − 1.38·75-s + 5/9·81-s + 0.878·83-s + 2.57·87-s − 3.31·93-s − 3.15·103-s − 0.383·109-s + 2.27·111-s + 3.38·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.15·147-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(56448\)    =    \(2^{7} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{56448} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 56448,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.148164301\)
\(L(\frac12)\) \(\approx\) \(2.148164301\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.656797755147918818405236990829, −9.541792988687802337799065273854, −9.172969029955001041869475015652, −8.228097915824748265448934137029, −8.173282349834764667604075089133, −7.33253412720106144652855023371, −7.26958585488936859387169650645, −6.30552876175851347232907272447, −5.77649722095232052155724795965, −5.01357758335939619070346679818, −4.41794336687255381402125784080, −3.60246495583988551500687864369, −3.14826068230388029805668446658, −2.34965244557392920909834502109, −1.35556028534757571668534423075, 1.35556028534757571668534423075, 2.34965244557392920909834502109, 3.14826068230388029805668446658, 3.60246495583988551500687864369, 4.41794336687255381402125784080, 5.01357758335939619070346679818, 5.77649722095232052155724795965, 6.30552876175851347232907272447, 7.26958585488936859387169650645, 7.33253412720106144652855023371, 8.173282349834764667604075089133, 8.228097915824748265448934137029, 9.172969029955001041869475015652, 9.541792988687802337799065273854, 9.656797755147918818405236990829

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