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Properties

Label 4-1184e2-1.1-c1e2-0-3
Degree $4$
Conductor $1401856$
Sign $1$
Analytic cond. $89.3835$
Root an. cond. $3.07478$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 2·7-s − 2·9-s − 5·11-s + 13-s + 15-s + 12·19-s + 2·21-s + 13·23-s − 6·25-s − 2·27-s + 17·29-s − 7·31-s − 5·33-s + 2·35-s − 2·37-s + 39-s − 3·41-s + 10·43-s − 2·45-s − 14·47-s + 2·49-s − 5·55-s + 12·57-s − 2·59-s + 23·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.755·7-s − 2/3·9-s − 1.50·11-s + 0.277·13-s + 0.258·15-s + 2.75·19-s + 0.436·21-s + 2.71·23-s − 6/5·25-s − 0.384·27-s + 3.15·29-s − 1.25·31-s − 0.870·33-s + 0.338·35-s − 0.328·37-s + 0.160·39-s − 0.468·41-s + 1.52·43-s − 0.298·45-s − 2.04·47-s + 2/7·49-s − 0.674·55-s + 1.58·57-s − 0.260·59-s + 2.94·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1401856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1401856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1401856\)    =    \(2^{10} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(89.3835\)
Root analytic conductor: \(3.07478\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1401856,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.366985453\)
\(L(\frac12)\) \(\approx\) \(3.366985453\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_d
5$D_{4}$ \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_h
7$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_c
11$D_{4}$ \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_z
13$D_{4}$ \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_x
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$D_{4}$ \( 1 - 13 T + 85 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.23.an_dh
29$D_{4}$ \( 1 - 17 T + 127 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.29.ar_ex
31$D_{4}$ \( 1 + 7 T + 71 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.31.h_ct
41$D_{4}$ \( 1 + 3 T + 81 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_dd
43$D_{4}$ \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_du
47$D_{4}$ \( 1 + 14 T + 130 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.47.o_fa
53$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.53.a_cc
59$D_{4}$ \( 1 + 2 T + 106 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_ec
61$D_{4}$ \( 1 - 23 T + 251 T^{2} - 23 p T^{3} + p^{2} T^{4} \) 2.61.ax_jr
67$D_{4}$ \( 1 + 5 T + 137 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.67.f_fh
71$D_{4}$ \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_ec
73$D_{4}$ \( 1 + 5 T + 149 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.73.f_ft
79$D_{4}$ \( 1 - 13 T + 197 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.79.an_hp
83$D_{4}$ \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.83.au_ig
89$D_{4}$ \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_gg
97$D_{4}$ \( 1 - 16 T + 206 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_hy
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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

−10.00416080521737867668547719186, −9.423573357901088040967922984435, −9.202591449590049307777214992322, −8.747654813880771261317206236437, −8.197551749731517472587885985878, −8.057561852321722246700430421064, −7.63426081872345604936301083114, −7.15973106300545898049087229000, −6.74102344402333627137106956186, −6.23015906320180027395520780820, −5.39965336207809609555544966994, −5.19785914764143636224463035061, −5.19272788969999737228410079259, −4.51811136958919140519267599375, −3.50297883050232801474336711920, −3.29981558819210314937142935625, −2.61111439831505776880749585905, −2.49515765356552428200768414999, −1.38697002448220497391236007967, −0.845839199601194662638915560879, 0.845839199601194662638915560879, 1.38697002448220497391236007967, 2.49515765356552428200768414999, 2.61111439831505776880749585905, 3.29981558819210314937142935625, 3.50297883050232801474336711920, 4.51811136958919140519267599375, 5.19272788969999737228410079259, 5.19785914764143636224463035061, 5.39965336207809609555544966994, 6.23015906320180027395520780820, 6.74102344402333627137106956186, 7.15973106300545898049087229000, 7.63426081872345604936301083114, 8.057561852321722246700430421064, 8.197551749731517472587885985878, 8.747654813880771261317206236437, 9.202591449590049307777214992322, 9.423573357901088040967922984435, 10.00416080521737867668547719186

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