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Properties

Label 4-3024e2-1.1-c1e2-0-31
Degree $4$
Conductor $9144576$
Sign $1$
Analytic cond. $583.066$
Root an. cond. $4.91393$
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  + 5-s − 2·7-s + 3·13-s + 2·17-s − 2·19-s + 5·23-s − 25-s + 7·29-s − 11·31-s − 2·35-s + 3·37-s − 7·41-s − 4·43-s + 5·47-s + 3·49-s + 7·53-s + 12·59-s + 14·61-s + 3·65-s − 15·67-s + 9·71-s + 20·73-s + 79-s − 3·83-s + 2·85-s + 11·89-s − 6·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 0.832·13-s + 0.485·17-s − 0.458·19-s + 1.04·23-s − 1/5·25-s + 1.29·29-s − 1.97·31-s − 0.338·35-s + 0.493·37-s − 1.09·41-s − 0.609·43-s + 0.729·47-s + 3/7·49-s + 0.961·53-s + 1.56·59-s + 1.79·61-s + 0.372·65-s − 1.83·67-s + 1.06·71-s + 2.34·73-s + 0.112·79-s − 0.329·83-s + 0.216·85-s + 1.16·89-s − 0.628·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(583.066\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.791876736\)
\(L(\frac12)\) \(\approx\) \(2.791876736\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_c
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$D_{4}$ \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_u
17$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.17.ac_bj
19$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_g
23$D_{4}$ \( 1 - 5 T + 44 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.23.af_bs
29$D_{4}$ \( 1 - 7 T + 62 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.29.ah_ck
31$D_{4}$ \( 1 + 11 T + 84 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.31.l_dg
37$D_{4}$ \( 1 - 3 T + 68 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.37.ad_cq
41$D_{4}$ \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.41.h_u
43$D_{4}$ \( 1 + 4 T + 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_cf
47$D_{4}$ \( 1 - 5 T + 92 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.47.af_do
53$D_{4}$ \( 1 - 7 T + 44 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.53.ah_bs
59$D_{4}$ \( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_er
61$D_{4}$ \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.61.ao_fi
67$D_{4}$ \( 1 + 15 T + 182 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.67.p_ha
71$D_{4}$ \( 1 - 9 T + 154 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.71.aj_fy
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$D_{4}$ \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_dg
83$D_{4}$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.83.d_abm
89$D_{4}$ \( 1 - 11 T + 200 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.89.al_hs
97$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_co
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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

−8.892718293254062412877276672571, −8.625822802351918334371111756075, −8.091978182629966504663112350471, −8.002859637729144642861460413568, −7.21814140709912005703809850466, −6.99039373219252135102644403294, −6.61993967489953020470612512418, −6.43085136583661267441879343046, −5.67563513701856352513182058358, −5.60968733392385391835693927802, −5.20451243606647193847976728981, −4.69430592600812520093669440916, −4.00660094813309536716252547278, −3.85677216269877334658197503783, −3.19403482852342196908607617564, −3.04556959614753020670151225230, −2.16296585064934691738795203471, −1.99310064607562990537866144910, −1.07238141199864987003897051941, −0.60315904700213309406701555834, 0.60315904700213309406701555834, 1.07238141199864987003897051941, 1.99310064607562990537866144910, 2.16296585064934691738795203471, 3.04556959614753020670151225230, 3.19403482852342196908607617564, 3.85677216269877334658197503783, 4.00660094813309536716252547278, 4.69430592600812520093669440916, 5.20451243606647193847976728981, 5.60968733392385391835693927802, 5.67563513701856352513182058358, 6.43085136583661267441879343046, 6.61993967489953020470612512418, 6.99039373219252135102644403294, 7.21814140709912005703809850466, 8.002859637729144642861460413568, 8.091978182629966504663112350471, 8.625822802351918334371111756075, 8.892718293254062412877276672571

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