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Properties

Label 4-221184-1.1-c1e2-0-27
Degree $4$
Conductor $221184$
Sign $-1$
Analytic cond. $14.1028$
Root an. cond. $1.93788$
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  + 3-s + 9-s − 4·11-s − 6·17-s − 6·25-s + 27-s − 4·33-s − 6·41-s + 12·43-s − 2·49-s − 6·51-s − 4·59-s − 4·73-s − 6·75-s + 81-s + 12·83-s − 30·89-s − 16·97-s − 4·99-s − 20·107-s + 18·113-s − 10·121-s − 6·123-s + 127-s + 12·129-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.45·17-s − 6/5·25-s + 0.192·27-s − 0.696·33-s − 0.937·41-s + 1.82·43-s − 2/7·49-s − 0.840·51-s − 0.520·59-s − 0.468·73-s − 0.692·75-s + 1/9·81-s + 1.31·83-s − 3.17·89-s − 1.62·97-s − 0.402·99-s − 1.93·107-s + 1.69·113-s − 0.909·121-s − 0.541·123-s + 0.0887·127-s + 1.05·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(221184\)    =    \(2^{13} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(14.1028\)
Root analytic conductor: \(1.93788\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 221184,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.g_bq
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.29.a_o
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.g_de
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.am_di
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.e_di
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.71.a_bi
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.e_g
79$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.79.a_abu
83$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.am_cg
89$C_2$$\times$$C_2$ \( ( 1 + 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.be_pm
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.q_io
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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

−8.890684117153239260936767635249, −8.160621380068486924542675300619, −8.005708205220096927606302337990, −7.43658575047987658029534243699, −6.93871798682740762921494233000, −6.46483533690380770441109142148, −5.78570120551876332611694119855, −5.38354531121501917965184121665, −4.67862249220530790151501692732, −4.21082653743022893223538445043, −3.65442284270181688974740888694, −2.74724914569650934525344512326, −2.43565109152823317355680712246, −1.58339733376362196907314613200, 0, 1.58339733376362196907314613200, 2.43565109152823317355680712246, 2.74724914569650934525344512326, 3.65442284270181688974740888694, 4.21082653743022893223538445043, 4.67862249220530790151501692732, 5.38354531121501917965184121665, 5.78570120551876332611694119855, 6.46483533690380770441109142148, 6.93871798682740762921494233000, 7.43658575047987658029534243699, 8.005708205220096927606302337990, 8.160621380068486924542675300619, 8.890684117153239260936767635249

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