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Properties

Label 4-2178e2-1.1-c1e2-0-29
Degree $4$
Conductor $4743684$
Sign $1$
Analytic cond. $302.461$
Root an. cond. $4.17030$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 5·5-s + 7-s − 4·8-s + 10·10-s − 2·13-s − 2·14-s + 5·16-s + 8·17-s − 2·19-s − 15·20-s − 2·23-s + 10·25-s + 4·26-s + 3·28-s + 7·29-s − 9·31-s − 6·32-s − 16·34-s − 5·35-s + 4·38-s + 20·40-s + 2·41-s + 14·43-s + 4·46-s − 8·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 2.23·5-s + 0.377·7-s − 1.41·8-s + 3.16·10-s − 0.554·13-s − 0.534·14-s + 5/4·16-s + 1.94·17-s − 0.458·19-s − 3.35·20-s − 0.417·23-s + 2·25-s + 0.784·26-s + 0.566·28-s + 1.29·29-s − 1.61·31-s − 1.06·32-s − 2.74·34-s − 0.845·35-s + 0.648·38-s + 3.16·40-s + 0.312·41-s + 2.13·43-s + 0.589·46-s − 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4743684\)    =    \(2^{2} \cdot 3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(302.461\)
Root analytic conductor: \(4.17030\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4743684,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
11 \( 1 \)
good5$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.5.f_p
7$D_{4}$ \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_n
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_w
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$C_4$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ag
23$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_c
29$D_{4}$ \( 1 - 7 T + 59 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.29.ah_ch
31$D_{4}$ \( 1 + 9 T + 81 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.31.j_dd
37$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.37.a_ag
41$D_{4}$ \( 1 - 2 T + 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_bm
43$D_{4}$ \( 1 - 14 T + 130 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.43.ao_fa
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.47.i_eg
53$D_{4}$ \( 1 + 5 T + 111 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.53.f_eh
59$D_{4}$ \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.59.f_cl
61$D_{4}$ \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_di
67$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_cs
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$D_{4}$ \( 1 + 9 T + 65 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.73.j_cn
79$D_{4}$ \( 1 - 7 T + 159 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.79.ah_gd
83$D_{4}$ \( 1 - T + 165 T^{2} - p T^{3} + p^{2} T^{4} \) 2.83.ab_gj
89$D_{4}$ \( 1 + 18 T + 254 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.89.s_ju
97$D_{4}$ \( 1 + 17 T + 265 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.97.r_kf
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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

−8.571398442519576546056692239195, −8.554249577586263838823781824123, −7.87702622378431927324424085616, −7.85155624215270234280771821053, −7.57386470102065008208215169188, −7.29400151503876532568419346440, −6.77639687071162104883806533466, −6.28675156989284991106500874141, −5.78055161833745351513169734293, −5.40683831816905627277194622741, −4.65910799135667981865031811685, −4.42561196718115706859328031311, −3.82544608105155173245846654480, −3.43120500942823317569811546868, −2.96628642418390958624647178880, −2.50782400879846964844710278097, −1.48912394830043730580450194365, −1.22839740464310038520551603811, 0, 0, 1.22839740464310038520551603811, 1.48912394830043730580450194365, 2.50782400879846964844710278097, 2.96628642418390958624647178880, 3.43120500942823317569811546868, 3.82544608105155173245846654480, 4.42561196718115706859328031311, 4.65910799135667981865031811685, 5.40683831816905627277194622741, 5.78055161833745351513169734293, 6.28675156989284991106500874141, 6.77639687071162104883806533466, 7.29400151503876532568419346440, 7.57386470102065008208215169188, 7.85155624215270234280771821053, 7.87702622378431927324424085616, 8.554249577586263838823781824123, 8.571398442519576546056692239195

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