Properties

Label 4-663552-1.1-c1e2-0-17
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·11-s − 2·17-s + 2·25-s − 2·41-s + 12·43-s − 10·49-s + 4·59-s + 16·67-s − 20·73-s + 12·83-s − 18·89-s + 4·107-s − 18·113-s + 6·121-s + ⋯
L(s)  = 1  − 1.20·11-s − 0.485·17-s + 2/5·25-s − 0.312·41-s + 1.82·43-s − 1.42·49-s + 0.520·59-s + 1.95·67-s − 2.34·73-s + 1.31·83-s − 1.90·89-s + 0.386·107-s − 1.69·113-s + 6/11·121-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: \(1\)
Selberg data: \((4,\ 663552,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.e_k
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.c_k
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.31.a_ao
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.c_de
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.43.am_eo
47$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.47.a_c
53$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.53.a_ck
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ae_di
61$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.61.a_bm
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.aq_ha
71$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.71.a_adq
73$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.u_iw
79$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.79.a_ady
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.s_gw
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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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

−8.174794089752187139499944114197, −7.72460189085224831079236942082, −7.19027694439500553096570616574, −6.91422618612056916135524716411, −6.24750675746707254480215507789, −5.88315392153081095098554167328, −5.27843592741937725955491607348, −4.96417388610455717524178965899, −4.37183033357980208209709299949, −3.86741899692196270821211658397, −3.15678748754127995417252398787, −2.62465335364738803754224841565, −2.12259302398545282902293463063, −1.15528183001150438942425843617, 0, 1.15528183001150438942425843617, 2.12259302398545282902293463063, 2.62465335364738803754224841565, 3.15678748754127995417252398787, 3.86741899692196270821211658397, 4.37183033357980208209709299949, 4.96417388610455717524178965899, 5.27843592741937725955491607348, 5.88315392153081095098554167328, 6.24750675746707254480215507789, 6.91422618612056916135524716411, 7.19027694439500553096570616574, 7.72460189085224831079236942082, 8.174794089752187139499944114197

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