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Properties

Label 4-5808e2-1.1-c1e2-0-8
Degree $4$
Conductor $33732864$
Sign $1$
Analytic cond. $2150.83$
Root an. cond. $6.81007$
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  + 2·3-s − 5-s + 4·7-s + 3·9-s + 8·13-s − 2·15-s − 7·17-s + 19-s + 8·21-s − 4·23-s + 2·25-s + 4·27-s + 15·31-s − 4·35-s − 12·37-s + 16·39-s − 10·41-s + 10·43-s − 3·45-s − 5·47-s + 3·49-s − 14·51-s − 7·53-s + 2·57-s − 11·59-s + 3·61-s + 12·63-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 1.51·7-s + 9-s + 2.21·13-s − 0.516·15-s − 1.69·17-s + 0.229·19-s + 1.74·21-s − 0.834·23-s + 2/5·25-s + 0.769·27-s + 2.69·31-s − 0.676·35-s − 1.97·37-s + 2.56·39-s − 1.56·41-s + 1.52·43-s − 0.447·45-s − 0.729·47-s + 3/7·49-s − 1.96·51-s − 0.961·53-s + 0.264·57-s − 1.43·59-s + 0.384·61-s + 1.51·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33732864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33732864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(33732864\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(2150.83\)
Root analytic conductor: \(6.81007\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 33732864,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.423260207\)
\(L(\frac12)\) \(\approx\) \(6.423260207\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good5$D_{4}$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ab
7$D_{4}$ \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_n
13$D_{4}$ \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.13.ai_bl
17$D_{4}$ \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.17.h_bt
19$D_{4}$ \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) 2.19.ab_h
23$D_{4}$ \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_bt
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$D_{4}$ \( 1 - 15 T + 117 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.31.ap_en
37$D_{4}$ \( 1 + 12 T + 105 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.37.m_eb
41$D_{4}$ \( 1 + 10 T + 87 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.41.k_dj
43$D_{4}$ \( 1 - 10 T + 91 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_dn
47$D_{4}$ \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.47.f_dv
53$D_{4}$ \( 1 + 7 T + 17 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.53.h_r
59$D_{4}$ \( 1 + 11 T + 137 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.59.l_fh
61$D_{4}$ \( 1 - 3 T + 123 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.61.ad_et
67$D_{4}$ \( 1 - 7 T + 135 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.67.ah_ff
71$D_{4}$ \( 1 - 5 T + 147 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.71.af_fr
73$D_{4}$ \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_fm
79$D_{4}$ \( 1 - 24 T + 297 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.79.ay_ll
83$D_{4}$ \( 1 - 22 T + 267 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.83.aw_kh
89$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.89.ac_gx
97$D_{4}$ \( 1 - T + 163 T^{2} - p T^{3} + p^{2} T^{4} \) 2.97.ab_gh
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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

−8.238701236636564451331677014662, −8.181182132872386575625230609059, −7.58852183466220987938246228417, −7.57127969934385176511766945908, −6.73495604490247376647518882382, −6.59299333708480298433216367895, −6.23604382288603337616885189336, −6.03253519832929230463042143898, −5.00716835657789161052398499596, −4.98826537686733075996705597320, −4.73923028091202161496117583029, −4.15985414936250733075534691836, −3.74569795789100316138439385073, −3.60818146771917266348490086330, −3.06569853628792248565410626049, −2.56111781369464597263741278414, −1.87320494791862447124187170199, −1.85844511111894555186141197849, −1.20503737985211782268765787417, −0.62974504540038199137308396807, 0.62974504540038199137308396807, 1.20503737985211782268765787417, 1.85844511111894555186141197849, 1.87320494791862447124187170199, 2.56111781369464597263741278414, 3.06569853628792248565410626049, 3.60818146771917266348490086330, 3.74569795789100316138439385073, 4.15985414936250733075534691836, 4.73923028091202161496117583029, 4.98826537686733075996705597320, 5.00716835657789161052398499596, 6.03253519832929230463042143898, 6.23604382288603337616885189336, 6.59299333708480298433216367895, 6.73495604490247376647518882382, 7.57127969934385176511766945908, 7.58852183466220987938246228417, 8.181182132872386575625230609059, 8.238701236636564451331677014662

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