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Properties

Label 4-2952e2-1.1-c1e2-0-2
Degree $4$
Conductor $8714304$
Sign $1$
Analytic cond. $555.631$
Root an. cond. $4.85508$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 6·11-s − 4·17-s + 6·19-s − 4·23-s − 10·25-s − 4·31-s − 8·37-s + 2·41-s − 2·47-s − 8·49-s − 12·59-s − 20·61-s + 2·67-s + 18·71-s − 4·73-s − 12·77-s − 2·79-s − 12·89-s − 12·97-s + 8·101-s − 12·103-s − 16·107-s − 12·113-s − 8·119-s + 8·121-s + 127-s + ⋯
L(s)  = 1  + 0.755·7-s − 1.80·11-s − 0.970·17-s + 1.37·19-s − 0.834·23-s − 2·25-s − 0.718·31-s − 1.31·37-s + 0.312·41-s − 0.291·47-s − 8/7·49-s − 1.56·59-s − 2.56·61-s + 0.244·67-s + 2.13·71-s − 0.468·73-s − 1.36·77-s − 0.225·79-s − 1.27·89-s − 1.21·97-s + 0.796·101-s − 1.18·103-s − 1.54·107-s − 1.12·113-s − 0.733·119-s + 8/11·121-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8714304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8714304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8714304\)    =    \(2^{6} \cdot 3^{4} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(555.631\)
Root analytic conductor: \(4.85508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8714304,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
41$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_m
11$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_bc
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$D_{4}$ \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_bs
23$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_bm
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.29.a_k
31$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_cc
37$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_bq
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.43.a_bm
47$D_{4}$ \( 1 + 2 T + 68 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_cq
53$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.53.a_cg
59$D_{4}$ \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_fm
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$D_{4}$ \( 1 - 2 T - 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_am
71$D_{4}$ \( 1 - 18 T + 220 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.71.as_im
73$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_bq
79$D_{4}$ \( 1 + 2 T + 84 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_dg
83$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \) 2.83.a_eo
89$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_gk
97$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_ha
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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

−8.286289158902572182289902597005, −8.126460458108805192496831856929, −7.77498802893600109353318019227, −7.71889437272887414334893527942, −7.09117416734023486395454444307, −6.77540324923524408941841832735, −6.19329602476583577444451698409, −5.78000824108561667544040083947, −5.40476158832847918717260241765, −5.22454643791634898247255066036, −4.52099448914007692858175848365, −4.51211203198845156269026667742, −3.61360768015275677510106840977, −3.48542660399747333286719307764, −2.67010244391774028355284122350, −2.45145455920581583429519532804, −1.65211041494787400296093141790, −1.52091643225347200116460795101, 0, 0, 1.52091643225347200116460795101, 1.65211041494787400296093141790, 2.45145455920581583429519532804, 2.67010244391774028355284122350, 3.48542660399747333286719307764, 3.61360768015275677510106840977, 4.51211203198845156269026667742, 4.52099448914007692858175848365, 5.22454643791634898247255066036, 5.40476158832847918717260241765, 5.78000824108561667544040083947, 6.19329602476583577444451698409, 6.77540324923524408941841832735, 7.09117416734023486395454444307, 7.71889437272887414334893527942, 7.77498802893600109353318019227, 8.126460458108805192496831856929, 8.286289158902572182289902597005

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