Properties

Label 4-44e4-1.1-c1e2-0-29
Degree $4$
Conductor $3748096$
Sign $1$
Analytic cond. $238.981$
Root an. cond. $3.93179$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 5-s − 7-s + 2·9-s − 7·13-s − 3·15-s − 7·17-s + 19-s − 3·21-s − 4·23-s − 8·25-s − 6·27-s − 15·29-s + 5·31-s + 35-s + 3·37-s − 21·39-s − 15·41-s − 2·45-s − 5·47-s − 2·49-s − 21·51-s + 3·53-s + 3·57-s + 9·59-s − 7·61-s − 2·63-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s − 0.377·7-s + 2/3·9-s − 1.94·13-s − 0.774·15-s − 1.69·17-s + 0.229·19-s − 0.654·21-s − 0.834·23-s − 8/5·25-s − 1.15·27-s − 2.78·29-s + 0.898·31-s + 0.169·35-s + 0.493·37-s − 3.36·39-s − 2.34·41-s − 0.298·45-s − 0.729·47-s − 2/7·49-s − 2.94·51-s + 0.412·53-s + 0.397·57-s + 1.17·59-s − 0.896·61-s − 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3748096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3748096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3748096\)    =    \(2^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(238.981\)
Root analytic conductor: \(3.93179\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3748096,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
11 \( 1 \)
good3$D_{4}$ \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.3.ad_h
5$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_j
7$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_d
13$D_{4}$ \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.13.h_bl
17$D_{4}$ \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.17.h_bt
19$D_{4}$ \( 1 - T + 27 T^{2} - p T^{3} + p^{2} T^{4} \) 2.19.ab_bb
23$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_be
29$D_{4}$ \( 1 + 15 T + 113 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.29.p_ej
31$D_{4}$ \( 1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.31.af_cf
37$D_{4}$ \( 1 - 3 T + 65 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.37.ad_cn
41$D_{4}$ \( 1 + 15 T + 137 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.41.p_fh
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$D_{4}$ \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.47.f_dv
53$D_{4}$ \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_cz
59$D_{4}$ \( 1 - 9 T + 107 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.59.aj_ed
61$D_{4}$ \( 1 + 7 T + 133 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_fd
67$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_cs
71$D_{4}$ \( 1 - 15 T + 197 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.71.ap_hp
73$D_{4}$ \( 1 - 13 T + 157 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.73.an_gb
79$D_{4}$ \( 1 + 21 T + 257 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.79.v_jx
83$D_{4}$ \( 1 + 3 T - 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.83.d_abr
89$D_{4}$ \( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.89.i_gs
97$D_{4}$ \( 1 + 9 T + 113 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.97.j_ej
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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

−8.907618555286815986152215308291, −8.606887356982895378010607812872, −8.171012528153261017282759886063, −8.013965104140292894914958310958, −7.37646976334079423506956314943, −7.34957434509773476134785267621, −6.78408496968783582273860281623, −6.36139680820959847046183636300, −5.59425958332330909946573147418, −5.54194021303638606515520693656, −4.73901643958458461400242365315, −4.36179112766088104105434483064, −3.90626743039437135573492606002, −3.43667875509020437934482439249, −3.09698048951235863539915812351, −2.46729303052924568707987870235, −1.96831034018598758301745213179, −1.95697076230749261092747981296, 0, 0, 1.95697076230749261092747981296, 1.96831034018598758301745213179, 2.46729303052924568707987870235, 3.09698048951235863539915812351, 3.43667875509020437934482439249, 3.90626743039437135573492606002, 4.36179112766088104105434483064, 4.73901643958458461400242365315, 5.54194021303638606515520693656, 5.59425958332330909946573147418, 6.36139680820959847046183636300, 6.78408496968783582273860281623, 7.34957434509773476134785267621, 7.37646976334079423506956314943, 8.013965104140292894914958310958, 8.171012528153261017282759886063, 8.606887356982895378010607812872, 8.907618555286815986152215308291

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