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Properties

Label 4-207e2-1.1-c1e2-0-4
Degree $4$
Conductor $42849$
Sign $1$
Analytic cond. $2.73208$
Root an. cond. $1.28565$
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  − 2·2-s + 4-s − 4·5-s − 4·7-s + 8·10-s + 8·14-s + 16-s − 12·17-s − 4·19-s − 4·20-s − 2·23-s + 4·25-s − 4·28-s + 2·32-s + 24·34-s + 16·35-s − 4·37-s + 8·38-s − 8·41-s − 12·43-s + 4·46-s + 12·47-s − 8·50-s − 4·53-s − 4·59-s + 4·61-s − 11·64-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s − 1.78·5-s − 1.51·7-s + 2.52·10-s + 2.13·14-s + 1/4·16-s − 2.91·17-s − 0.917·19-s − 0.894·20-s − 0.417·23-s + 4/5·25-s − 0.755·28-s + 0.353·32-s + 4.11·34-s + 2.70·35-s − 0.657·37-s + 1.29·38-s − 1.24·41-s − 1.82·43-s + 0.589·46-s + 1.75·47-s − 1.13·50-s − 0.549·53-s − 0.520·59-s + 0.512·61-s − 1.37·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42849 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42849 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(42849\)    =    \(3^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(2.73208\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 42849,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3 \( 1 \)
23$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.2.c_d
5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_m
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_q
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$D_{4}$ \( 1 + 12 T + 4 p T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.17.m_cq
19$D_{4}$ \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_y
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.29.a_ao
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.31.a_ak
37$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_cs
41$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_co
43$D_{4}$ \( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_ea
47$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_du
53$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_ci
59$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_dm
61$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_eo
67$D_{4}$ \( 1 - 20 T + 232 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.67.au_iy
71$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.71.aq_gs
73$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.73.ae_w
79$D_{4}$ \( 1 + 4 T + 64 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.79.e_cm
83$D_{4}$ \( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_gs
89$D_{4}$ \( 1 + 12 T + 164 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_gi
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_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

−12.04306302795660426804772859266, −11.52713216980327215340107252533, −11.09915204036058115924477961631, −10.71860312590790766445805174303, −9.964339978482274388033449522212, −9.693209197077215297876867136019, −9.064007541260421417540531878023, −8.698348566219021670728500586163, −8.173836918809145473864207718720, −8.043276254295977733764515035215, −6.87087535512029452118997036170, −6.82579595939579436685762391846, −6.36460821236070883670931088360, −5.23283621324211346579253115359, −4.27328667449444397289866024716, −3.98853768470853157072991182226, −3.21105387653454934310381798050, −2.20177691065970790790432842988, 0, 0, 2.20177691065970790790432842988, 3.21105387653454934310381798050, 3.98853768470853157072991182226, 4.27328667449444397289866024716, 5.23283621324211346579253115359, 6.36460821236070883670931088360, 6.82579595939579436685762391846, 6.87087535512029452118997036170, 8.043276254295977733764515035215, 8.173836918809145473864207718720, 8.698348566219021670728500586163, 9.064007541260421417540531878023, 9.693209197077215297876867136019, 9.964339978482274388033449522212, 10.71860312590790766445805174303, 11.09915204036058115924477961631, 11.52713216980327215340107252533, 12.04306302795660426804772859266

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