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Properties

Label 4-272e2-1.1-c1e2-0-5
Degree $4$
Conductor $73984$
Sign $1$
Analytic cond. $4.71728$
Root an. cond. $1.47374$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·7-s + 6·11-s + 4·13-s − 2·17-s − 4·19-s − 4·21-s + 6·23-s + 2·25-s + 2·27-s + 2·31-s − 12·33-s + 16·37-s − 8·39-s − 12·41-s − 4·43-s − 8·49-s + 4·51-s + 12·53-s + 8·57-s − 12·59-s − 8·61-s − 16·67-s − 12·69-s + 6·71-s + 4·73-s − 4·75-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.755·7-s + 1.80·11-s + 1.10·13-s − 0.485·17-s − 0.917·19-s − 0.872·21-s + 1.25·23-s + 2/5·25-s + 0.384·27-s + 0.359·31-s − 2.08·33-s + 2.63·37-s − 1.28·39-s − 1.87·41-s − 0.609·43-s − 8/7·49-s + 0.560·51-s + 1.64·53-s + 1.05·57-s − 1.56·59-s − 1.02·61-s − 1.95·67-s − 1.44·69-s + 0.712·71-s + 0.468·73-s − 0.461·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(73984\)    =    \(2^{8} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(4.71728\)
Root analytic conductor: \(1.47374\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 73984,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.159849312\)
\(L(\frac12)\) \(\approx\) \(1.159849312\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_m
11$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_bc
13$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_s
19$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_be
23$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_ca
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.29.a_bu
31$D_{4}$ \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_bk
37$D_{4}$ \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.37.aq_ew
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_as
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.47.a_bu
53$D_{4}$ \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_dq
59$D_{4}$ \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_fm
61$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_ew
67$D_{4}$ \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.67.q_fu
71$D_{4}$ \( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_eu
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$D_{4}$ \( 1 - 14 T + 180 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.79.ao_gy
83$D_{4}$ \( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_hi
89$D_{4}$ \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_hu
97$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_fu
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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

−11.90851821493114120444474468405, −11.74987426934830990023118406207, −11.10008005049718922968096856752, −10.90450427833972764767176417418, −10.67363771435516557663738383710, −9.714583208713393157827921763093, −9.235918519520059667721385490356, −8.866386016509338118300441791823, −8.305149448580549842280670425934, −7.904128954404987841992012299510, −6.92837634457615271208678548628, −6.61656676698846903378440893615, −6.12593233406656902593840342794, −5.81248361066555606837246640663, −4.75751406190003228425980836964, −4.68957200929919856795552314801, −3.83262757244240793968393380066, −3.10397805676277836343171329090, −1.84590143985780086612851937706, −0.989417793767317894634255709039, 0.989417793767317894634255709039, 1.84590143985780086612851937706, 3.10397805676277836343171329090, 3.83262757244240793968393380066, 4.68957200929919856795552314801, 4.75751406190003228425980836964, 5.81248361066555606837246640663, 6.12593233406656902593840342794, 6.61656676698846903378440893615, 6.92837634457615271208678548628, 7.904128954404987841992012299510, 8.305149448580549842280670425934, 8.866386016509338118300441791823, 9.235918519520059667721385490356, 9.714583208713393157827921763093, 10.67363771435516557663738383710, 10.90450427833972764767176417418, 11.10008005049718922968096856752, 11.74987426934830990023118406207, 11.90851821493114120444474468405

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