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Properties

Label 4-42e4-1.1-c1e2-0-3
Degree $4$
Conductor $3111696$
Sign $1$
Analytic cond. $198.404$
Root an. cond. $3.75308$
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  − 3·5-s − 3·11-s − 4·13-s − 3·17-s − 19-s + 3·23-s + 5·25-s + 12·29-s − 7·31-s + 37-s + 12·41-s − 8·43-s + 9·47-s + 3·53-s + 9·55-s − 9·59-s − 61-s + 12·65-s + 7·67-s − 73-s + 13·79-s + 24·83-s + 9·85-s − 15·89-s + 3·95-s + 20·97-s − 15·101-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.904·11-s − 1.10·13-s − 0.727·17-s − 0.229·19-s + 0.625·23-s + 25-s + 2.22·29-s − 1.25·31-s + 0.164·37-s + 1.87·41-s − 1.21·43-s + 1.31·47-s + 0.412·53-s + 1.21·55-s − 1.17·59-s − 0.128·61-s + 1.48·65-s + 0.855·67-s − 0.117·73-s + 1.46·79-s + 2.63·83-s + 0.976·85-s − 1.58·89-s + 0.307·95-s + 2.03·97-s − 1.49·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(198.404\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3111696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8187836127\)
\(L(\frac12)\) \(\approx\) \(0.8187836127\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_e
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.b_as
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.h_s
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.ab_abk
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_bi
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_abs
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.59.j_w
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.b_aci
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.67.ah_as
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) 2.73.b_acu
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.an_dm
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2^2$ \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.89.p_fg
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−9.275993192494616111285028642311, −9.126592344308872936867435904246, −8.687065203536388825834989353791, −8.231753682213813063892496311277, −7.88491011632925528730026328762, −7.52057771591072305014124119561, −7.26473740922889085785367969412, −6.86130523218797883858007644754, −6.24281146049264502037391873351, −6.06474461249329583708042060231, −5.04289591535035546902652983735, −5.03699017945427321285348744946, −4.69151708123459202889501143222, −4.02173828309510198941301179338, −3.70995277526017633003680313193, −3.08129912057261977144558726871, −2.42099977521082369946778193307, −2.37441098363492605969234423041, −1.13677278297615577680817401978, −0.37839860505665914040816438767, 0.37839860505665914040816438767, 1.13677278297615577680817401978, 2.37441098363492605969234423041, 2.42099977521082369946778193307, 3.08129912057261977144558726871, 3.70995277526017633003680313193, 4.02173828309510198941301179338, 4.69151708123459202889501143222, 5.03699017945427321285348744946, 5.04289591535035546902652983735, 6.06474461249329583708042060231, 6.24281146049264502037391873351, 6.86130523218797883858007644754, 7.26473740922889085785367969412, 7.52057771591072305014124119561, 7.88491011632925528730026328762, 8.231753682213813063892496311277, 8.687065203536388825834989353791, 9.126592344308872936867435904246, 9.275993192494616111285028642311

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