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Properties

Label 4-272e2-1.1-c1e2-0-27
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 + 4·5-s − 2·7-s + 2·9-s − 2·11-s + 8·15-s + 2·17-s + 4·19-s − 4·21-s − 2·23-s + 2·25-s + 6·27-s + 4·29-s + 2·31-s − 4·33-s − 8·35-s − 4·37-s + 4·41-s + 12·43-s + 8·45-s − 8·47-s − 6·49-s + 4·51-s − 4·53-s − 8·55-s + 8·57-s − 20·59-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s − 0.755·7-s + 2/3·9-s − 0.603·11-s + 2.06·15-s + 0.485·17-s + 0.917·19-s − 0.872·21-s − 0.417·23-s + 2/5·25-s + 1.15·27-s + 0.742·29-s + 0.359·31-s − 0.696·33-s − 1.35·35-s − 0.657·37-s + 0.624·41-s + 1.82·43-s + 1.19·45-s − 1.16·47-s − 6/7·49-s + 0.560·51-s − 0.549·53-s − 1.07·55-s + 1.05·57-s − 2.60·59-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

\(L(1)\) \(\approx\) \(2.677731571\)
\(L(\frac12)\) \(\approx\) \(2.677731571\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
7$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_k
11$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_s
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
19$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_w
23$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bq
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$D_{4}$ \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_cg
37$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_ac
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.43.am_dy
47$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_be
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$D_{4}$ \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.59.u_hq
61$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_bu
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$D_{4}$ \( 1 + 14 T + 186 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.71.o_he
73$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.73.am_dy
79$D_{4}$ \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_fi
83$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_ha
89$D_{4}$ \( 1 + 24 T + 302 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.89.y_lq
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−12.48856928227859852307148802935, −11.67358976323158480949015852791, −11.13712097916083713944755017265, −10.37247261907585513200028846242, −10.09932288963077603501628345531, −9.778384238222595061467140633825, −9.256892817145036237099022650492, −9.117821161770955834747579532639, −8.114884718896373013616156203111, −8.074541144363053444368778372887, −7.26532214798757090733578192429, −6.69345499335150516305026557026, −6.01681727017416252108927888744, −5.80387098886009721381694034758, −5.04509368732520633281911845185, −4.36771464994201450096785652791, −3.32584481261578245160488299107, −2.93544302415270856064158848348, −2.29602170294392146313536467747, −1.45395524885242083018695564650, 1.45395524885242083018695564650, 2.29602170294392146313536467747, 2.93544302415270856064158848348, 3.32584481261578245160488299107, 4.36771464994201450096785652791, 5.04509368732520633281911845185, 5.80387098886009721381694034758, 6.01681727017416252108927888744, 6.69345499335150516305026557026, 7.26532214798757090733578192429, 8.074541144363053444368778372887, 8.114884718896373013616156203111, 9.117821161770955834747579532639, 9.256892817145036237099022650492, 9.778384238222595061467140633825, 10.09932288963077603501628345531, 10.37247261907585513200028846242, 11.13712097916083713944755017265, 11.67358976323158480949015852791, 12.48856928227859852307148802935

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