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Properties

Label 4-5408e2-1.1-c1e2-0-2
Degree $4$
Conductor $29246464$
Sign $1$
Analytic cond. $1864.77$
Root an. cond. $6.57138$
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  + 2·7-s − 6·9-s − 6·11-s − 10·19-s − 10·25-s + 14·31-s + 18·47-s + 2·49-s − 4·53-s + 2·59-s + 12·61-s − 12·63-s + 22·67-s − 10·71-s − 12·77-s + 27·81-s + 14·83-s + 36·99-s − 28·101-s + 18·121-s + 127-s + 131-s − 20·133-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.755·7-s − 2·9-s − 1.80·11-s − 2.29·19-s − 2·25-s + 2.51·31-s + 2.62·47-s + 2/7·49-s − 0.549·53-s + 0.260·59-s + 1.53·61-s − 1.51·63-s + 2.68·67-s − 1.18·71-s − 1.36·77-s + 3·81-s + 1.53·83-s + 3.61·99-s − 2.78·101-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s − 1.73·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29246464\)    =    \(2^{10} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1864.77\)
Root analytic conductor: \(6.57138\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 29246464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9121815655\)
\(L(\frac12)\) \(\approx\) \(0.9121815655\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_c
11$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_s
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.19.k_by
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.31.ao_du
37$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.37.a_cw
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.47.as_gg
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.59.ac_c
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2^2$ \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.67.aw_ji
71$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.71.k_by
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.83.ao_du
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.97.a_hm
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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

−8.290920337103433968638082112083, −8.054887254600118427465542165919, −7.85843313138418233801110221546, −7.48814495962168796197319659406, −6.68683156605006657232244898791, −6.68610314191711379422290594135, −5.97965496253258053694474589459, −5.89966678642302658845833196152, −5.39873577745571645868623580820, −5.30230749210438140342246901954, −4.54386949014218662447282705892, −4.53968949114527022381611310034, −3.81973981670672599847778896931, −3.60470616381871799865439070652, −2.67759063260486464183517944594, −2.61419803635471967063127913195, −2.30943975288190432351896170059, −1.90677032215893998120061257157, −0.846203761821719565463581774344, −0.30369910279042619929670735439, 0.30369910279042619929670735439, 0.846203761821719565463581774344, 1.90677032215893998120061257157, 2.30943975288190432351896170059, 2.61419803635471967063127913195, 2.67759063260486464183517944594, 3.60470616381871799865439070652, 3.81973981670672599847778896931, 4.53968949114527022381611310034, 4.54386949014218662447282705892, 5.30230749210438140342246901954, 5.39873577745571645868623580820, 5.89966678642302658845833196152, 5.97965496253258053694474589459, 6.68610314191711379422290594135, 6.68683156605006657232244898791, 7.48814495962168796197319659406, 7.85843313138418233801110221546, 8.054887254600118427465542165919, 8.290920337103433968638082112083

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