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Properties

Label 4-342e2-1.1-c1e2-0-13
Degree $4$
Conductor $116964$
Sign $1$
Analytic cond. $7.45772$
Root an. cond. $1.65253$
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 + 2·7-s + 8-s + 4·11-s + 3·13-s − 2·14-s − 16-s + 4·17-s + 8·19-s − 4·22-s + 4·23-s + 5·25-s − 3·26-s − 6·31-s − 4·34-s − 10·37-s − 8·38-s + 4·41-s + 9·43-s − 4·46-s + 10·47-s − 11·49-s − 5·50-s − 4·53-s + 2·56-s − 14·59-s − 11·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.755·7-s + 0.353·8-s + 1.20·11-s + 0.832·13-s − 0.534·14-s − 1/4·16-s + 0.970·17-s + 1.83·19-s − 0.852·22-s + 0.834·23-s + 25-s − 0.588·26-s − 1.07·31-s − 0.685·34-s − 1.64·37-s − 1.29·38-s + 0.624·41-s + 1.37·43-s − 0.589·46-s + 1.45·47-s − 1.57·49-s − 0.707·50-s − 0.549·53-s + 0.267·56-s − 1.82·59-s − 1.40·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116964 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116964 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(116964\)    =    \(2^{2} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(7.45772\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 116964,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.428549112\)
\(L(\frac12)\) \(\approx\) \(1.428549112\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.7.ac_p
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_ae
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_ab
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_ah
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.29.a_abd
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.37.k_dv
41$C_2^2$ \( 1 - 4 T - 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_az
43$C_2^2$ \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.43.aj_bm
47$C_2^2$ \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_cb
53$C_2^2$ \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_abl
59$C_2^2$ \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.59.o_fh
61$C_2^2$ \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.61.l_ci
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.67.d_acg
71$C_2^2$ \( 1 - 14 T + 125 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.71.ao_ev
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.73.al_bw
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) 2.79.b_ada
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.83.q_iw
89$C_2^2$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.89.o_ed
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_adp
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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

−11.54348307164942019598761102173, −11.25716029823693530957412458319, −10.73515255721021597639373978246, −10.58374644622505703347783348294, −9.629396660823084285906428724082, −9.418110536988220635410834132230, −9.034949659383026950616417339883, −8.639031928671348426855592035890, −7.88499860384502907608527076965, −7.70689965926038983256584618083, −7.05318714727489721208141843972, −6.66304521052953388567010465269, −5.84833753542444496259914943911, −5.37755673532302851801213209369, −4.87586858957521118499595655105, −4.10415229931038676578117562284, −3.48617559706665704132131698909, −2.89243811900855163697293673280, −1.44523947780977611155881925702, −1.22373116820263713944100213477, 1.22373116820263713944100213477, 1.44523947780977611155881925702, 2.89243811900855163697293673280, 3.48617559706665704132131698909, 4.10415229931038676578117562284, 4.87586858957521118499595655105, 5.37755673532302851801213209369, 5.84833753542444496259914943911, 6.66304521052953388567010465269, 7.05318714727489721208141843972, 7.70689965926038983256584618083, 7.88499860384502907608527076965, 8.639031928671348426855592035890, 9.034949659383026950616417339883, 9.418110536988220635410834132230, 9.629396660823084285906428724082, 10.58374644622505703347783348294, 10.73515255721021597639373978246, 11.25716029823693530957412458319, 11.54348307164942019598761102173

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