Properties

Label 4-663552-1.1-c1e2-0-5
Degree $4$
Conductor $663552$
Sign $1$
Analytic cond. $42.3086$
Root an. cond. $2.55039$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·11-s + 4·17-s + 12·19-s + 2·25-s + 4·41-s − 12·43-s + 2·49-s + 4·59-s + 4·67-s + 4·73-s − 12·83-s + 12·89-s − 12·97-s + 4·107-s + 12·113-s − 6·121-s + ⋯
L(s)  = 1  − 1.20·11-s + 0.970·17-s + 2.75·19-s + 2/5·25-s + 0.624·41-s − 1.82·43-s + 2/7·49-s + 0.520·59-s + 0.488·67-s + 0.468·73-s − 1.31·83-s + 1.27·89-s − 1.21·97-s + 0.386·107-s + 1.12·113-s − 0.545·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663552 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663552 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(663552\)    =    \(2^{13} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(42.3086\)
Root analytic conductor: \(2.55039\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 663552,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.957343141\)
\(L(\frac12)\) \(\approx\) \(1.957343141\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.e_w
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ae_w
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.19.am_cs
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.29.a_abi
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.37.a_acg
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ae_cs
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.m_eo
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.47.a_abi
53$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.53.a_o
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ae_di
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.61.a_ak
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.ae_bm
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.71.a_ck
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ae_di
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.m_gk
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.m_gk
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.208794595534492784695142317876, −7.907414065336892021642106880479, −7.48358565736890846987585079470, −7.11721362355978233355219715987, −6.68610512203144024658459509195, −5.83682115782045162608821333991, −5.63161830591484917361607967688, −5.05545012848669388077222729486, −4.91553098075224197485797657207, −4.02540791711508105876008742301, −3.27413263309842694910341096922, −3.14891943782587258730301428040, −2.43335794389110709405349538329, −1.51448724696844858715127440409, −0.74175994915454536044766655904, 0.74175994915454536044766655904, 1.51448724696844858715127440409, 2.43335794389110709405349538329, 3.14891943782587258730301428040, 3.27413263309842694910341096922, 4.02540791711508105876008742301, 4.91553098075224197485797657207, 5.05545012848669388077222729486, 5.63161830591484917361607967688, 5.83682115782045162608821333991, 6.68610512203144024658459509195, 7.11721362355978233355219715987, 7.48358565736890846987585079470, 7.907414065336892021642106880479, 8.208794595534492784695142317876

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