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Properties

Label 4-3024e2-1.1-c1e2-0-14
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 5·7-s − 6·11-s + 13-s + 3·17-s − 7·19-s − 18·23-s + 17·25-s + 3·29-s + 8·31-s + 30·35-s + 37-s + 3·41-s − 43-s + 18·49-s + 3·53-s + 36·55-s − 2·61-s − 6·65-s − 4·67-s + 24·71-s − 11·73-s + 30·77-s − 16·79-s + 9·83-s − 18·85-s + 3·89-s + ⋯
L(s)  = 1  − 2.68·5-s − 1.88·7-s − 1.80·11-s + 0.277·13-s + 0.727·17-s − 1.60·19-s − 3.75·23-s + 17/5·25-s + 0.557·29-s + 1.43·31-s + 5.07·35-s + 0.164·37-s + 0.468·41-s − 0.152·43-s + 18/7·49-s + 0.412·53-s + 4.85·55-s − 0.256·61-s − 0.744·65-s − 0.488·67-s + 2.84·71-s − 1.28·73-s + 3.41·77-s − 1.80·79-s + 0.987·83-s − 1.95·85-s + 0.317·89-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

\(L(1)\) \(\approx\) \(0.3535492229\)
\(L(\frac12)\) \(\approx\) \(0.3535492229\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_am
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.h_be
23$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.23.s_ex
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_au
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.31.ai_bh
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.ab_abk
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_abg
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_abq
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_abs
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_acf
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_abz
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.73.l_bw
79$C_2^2$ \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.79.q_gv
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2^2$ \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.89.ad_adc
97$C_2^2$ \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) 2.97.ab_ads
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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

−8.517885854922948556694070336518, −8.314956312881759123224476564979, −8.174890139527913313774107040977, −7.974489177529554876701882570338, −7.41477524565974534704403796274, −7.23354629692893629091474083984, −6.69762086587224616322337897264, −6.21886383245703373889475541072, −5.92720732385006968741406965400, −5.70727236726681934865792631524, −4.83751325073459044826877946288, −4.43031001848698126545053107556, −3.99821846522307815142327887860, −3.97687565556806584445415874233, −3.24066762113550097094263412158, −3.15928926813210391127594788224, −2.42744002555340840886467671729, −2.05884008072415760072867498624, −0.52740392271826969054027773658, −0.37305863137397643257563003924, 0.37305863137397643257563003924, 0.52740392271826969054027773658, 2.05884008072415760072867498624, 2.42744002555340840886467671729, 3.15928926813210391127594788224, 3.24066762113550097094263412158, 3.97687565556806584445415874233, 3.99821846522307815142327887860, 4.43031001848698126545053107556, 4.83751325073459044826877946288, 5.70727236726681934865792631524, 5.92720732385006968741406965400, 6.21886383245703373889475541072, 6.69762086587224616322337897264, 7.23354629692893629091474083984, 7.41477524565974534704403796274, 7.974489177529554876701882570338, 8.174890139527913313774107040977, 8.314956312881759123224476564979, 8.517885854922948556694070336518

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