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Properties

Label 4-2592e2-1.1-c1e2-0-39
Degree $4$
Conductor $6718464$
Sign $1$
Analytic cond. $428.375$
Root an. cond. $4.54942$
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  + 2·5-s − 2·7-s − 2·11-s + 2·13-s − 8·19-s − 6·23-s − 7·25-s − 10·29-s − 10·31-s − 4·35-s + 8·37-s + 14·41-s − 10·43-s + 2·47-s − 5·49-s − 8·53-s − 4·55-s + 14·59-s − 6·61-s + 4·65-s − 10·67-s − 4·71-s + 4·77-s − 22·79-s − 6·83-s + 16·89-s − 4·91-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s − 0.603·11-s + 0.554·13-s − 1.83·19-s − 1.25·23-s − 7/5·25-s − 1.85·29-s − 1.79·31-s − 0.676·35-s + 1.31·37-s + 2.18·41-s − 1.52·43-s + 0.291·47-s − 5/7·49-s − 1.09·53-s − 0.539·55-s + 1.82·59-s − 0.768·61-s + 0.496·65-s − 1.22·67-s − 0.474·71-s + 0.455·77-s − 2.47·79-s − 0.658·83-s + 1.69·89-s − 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6718464\)    =    \(2^{10} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(428.375\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6718464,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
7$D_{4}$ \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_j
11$D_{4}$ \( 1 + 2 T + 17 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_r
13$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_d
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$D_{4}$ \( 1 + 6 T + 49 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_bx
29$D_{4}$ \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_ch
31$D_{4}$ \( 1 + 10 T + 81 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_dd
37$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_co
41$D_{4}$ \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.41.ao_ed
43$C_2^2$ \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_cf
47$D_{4}$ \( 1 - 2 T + 41 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.47.ac_bp
53$D_{4}$ \( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_du
59$D_{4}$ \( 1 - 14 T + 113 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_ej
61$D_{4}$ \( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_ed
67$D_{4}$ \( 1 + 10 T + 105 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_eb
71$D_{4}$ \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_by
73$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \) 2.73.a_es
79$D_{4}$ \( 1 + 22 T + 273 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.79.w_kn
83$D_{4}$ \( 1 + 6 T + 169 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_gn
89$D_{4}$ \( 1 - 16 T + 218 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.89.aq_ik
97$D_{4}$ \( 1 - 2 T + 171 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_gp
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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

−8.686143532857793241791236250634, −8.379692822897964880801089250395, −7.79544974870495292926968090564, −7.67665961244998519006654619672, −7.23141850394141235153283021534, −6.61075658676609055992475768597, −6.29397758324642813278496762442, −5.92570886775611348466188395512, −5.70575516029211423996731527898, −5.46339672204947388332563283709, −4.65740300593575659417585676423, −4.22746853875713388952674541519, −3.77381711099031049196490130231, −3.62349569499463121812328773839, −2.64610357926533492370237030524, −2.48460296539691962767286150289, −1.73490294330663327596617513969, −1.59855439628151168943818897603, 0, 0, 1.59855439628151168943818897603, 1.73490294330663327596617513969, 2.48460296539691962767286150289, 2.64610357926533492370237030524, 3.62349569499463121812328773839, 3.77381711099031049196490130231, 4.22746853875713388952674541519, 4.65740300593575659417585676423, 5.46339672204947388332563283709, 5.70575516029211423996731527898, 5.92570886775611348466188395512, 6.29397758324642813278496762442, 6.61075658676609055992475768597, 7.23141850394141235153283021534, 7.67665961244998519006654619672, 7.79544974870495292926968090564, 8.379692822897964880801089250395, 8.686143532857793241791236250634

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