Properties

Label 4-378e2-1.1-c1e2-0-2
Degree $4$
Conductor $142884$
Sign $1$
Analytic cond. $9.11040$
Root an. cond. $1.73733$
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-s − 4·7-s − 8-s − 6·11-s + 10·13-s − 4·14-s − 16-s − 6·17-s + 4·19-s − 6·22-s − 6·23-s + 5·25-s + 10·26-s + 12·29-s + 31-s − 6·34-s + 37-s + 4·38-s − 12·41-s − 2·43-s − 6·46-s + 6·47-s + 9·49-s + 5·50-s + 6·53-s + 4·56-s + 12·58-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.51·7-s − 0.353·8-s − 1.80·11-s + 2.77·13-s − 1.06·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s − 1.27·22-s − 1.25·23-s + 25-s + 1.96·26-s + 2.22·29-s + 0.179·31-s − 1.02·34-s + 0.164·37-s + 0.648·38-s − 1.87·41-s − 0.304·43-s − 0.884·46-s + 0.875·47-s + 9/7·49-s + 0.707·50-s + 0.824·53-s + 0.534·56-s + 1.57·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(142884\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(9.11040\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 142884,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.623240614\)
\(L(\frac12)\) \(\approx\) \(1.623240614\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_z
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.13.ak_bz
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_t
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_ad
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_n
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) 2.31.ab_abe
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.ab_abk
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_al
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ar
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ax
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.ab_aci
67$C_2^2$ \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \) 2.67.ab_aco
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_acr
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ada
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \) 2.97.abi_sp
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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

−12.05602328436749953487507509593, −11.08203437333440760215267638765, −10.57837394671706512395657219525, −10.38805947456277925055209132771, −9.918004128200334898047092817731, −9.285271695669495452058939774900, −8.645058975598650243652514653308, −8.470900719350957092343565843175, −8.061332521392267183355566815387, −7.14638130280981763896625698944, −6.65944530890540019030159135597, −6.13183919178818791994689393089, −6.08706715299460033565153939227, −5.09452636472162406048197616185, −4.87961467951190439939380869548, −3.78199205398717790541895117723, −3.64707567206504421166713519353, −2.91172864827382921688936916286, −2.30769262666340611120845357023, −0.77667562379779309584132223030, 0.77667562379779309584132223030, 2.30769262666340611120845357023, 2.91172864827382921688936916286, 3.64707567206504421166713519353, 3.78199205398717790541895117723, 4.87961467951190439939380869548, 5.09452636472162406048197616185, 6.08706715299460033565153939227, 6.13183919178818791994689393089, 6.65944530890540019030159135597, 7.14638130280981763896625698944, 8.061332521392267183355566815387, 8.470900719350957092343565843175, 8.645058975598650243652514653308, 9.285271695669495452058939774900, 9.918004128200334898047092817731, 10.38805947456277925055209132771, 10.57837394671706512395657219525, 11.08203437333440760215267638765, 12.05602328436749953487507509593

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