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Properties

Label 4-5200e2-1.1-c1e2-0-1
Degree $4$
Conductor $27040000$
Sign $1$
Analytic cond. $1724.09$
Root an. cond. $6.44377$
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-s − 7-s + 3·9-s + 3·11-s + 2·13-s − 3·17-s + 8·19-s − 21-s − 3·23-s + 8·27-s − 31-s + 3·33-s − 20·37-s + 2·39-s + 6·41-s − 13·43-s − 15·47-s − 5·49-s − 3·51-s − 12·53-s + 8·57-s − 3·59-s − 8·61-s − 3·63-s − 67-s − 3·69-s − 18·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 9-s + 0.904·11-s + 0.554·13-s − 0.727·17-s + 1.83·19-s − 0.218·21-s − 0.625·23-s + 1.53·27-s − 0.179·31-s + 0.522·33-s − 3.28·37-s + 0.320·39-s + 0.937·41-s − 1.98·43-s − 2.18·47-s − 5/7·49-s − 0.420·51-s − 1.64·53-s + 1.05·57-s − 0.390·59-s − 1.02·61-s − 0.377·63-s − 0.122·67-s − 0.361·69-s − 2.13·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27040000\)    =    \(2^{8} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1724.09\)
Root analytic conductor: \(6.44377\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 27040000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.410259429\)
\(L(\frac12)\) \(\approx\) \(2.410259429\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_ac
7$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_g
11$D_{4}$ \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_q
17$D_{4}$ \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_bc
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$D_{4}$ \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_bo
29$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.29.a_z
31$D_{4}$ \( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} \) 2.31.b_cc
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$D_{4}$ \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_cg
43$D_{4}$ \( 1 + 13 T + 120 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.43.n_eq
47$D_{4}$ \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.47.p_fm
53$D_{4}$ \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_ef
59$D_{4}$ \( 1 + 3 T + 112 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.59.d_ei
61$D_{4}$ \( 1 + 8 T + 105 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_eb
67$D_{4}$ \( 1 + T + 60 T^{2} + p T^{3} + p^{2} T^{4} \) 2.67.b_ci
71$D_{4}$ \( 1 + 18 T + 190 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.71.s_hi
73$D_{4}$ \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_ek
79$D_{4}$ \( 1 + 19 T + 240 T^{2} + 19 p T^{3} + p^{2} T^{4} \) 2.79.t_jg
83$D_{4}$ \( 1 + 21 T + 268 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.83.v_ki
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.89.ay_mk
97$D_{4}$ \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_ew
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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

−8.511440155065986906022627686112, −8.084053123728582976026940773758, −7.63729536031500328498681463323, −7.24758293111981625403311020988, −6.96476330116179416107196007825, −6.68841949709193800757686584592, −6.38280924394720604513236859033, −5.84007812889460138407102925505, −5.62028951728929534403938904447, −4.84661700716756271430388799358, −4.70918558880949836492788608586, −4.46710771200692897097441782903, −3.68338034009419020565675148433, −3.48574509724151396245479699417, −3.02839644430181302316841523805, −3.00090254452484019844807963947, −1.77508943274160610130662980499, −1.67262753856399767295504209267, −1.42597084608011824219471436529, −0.38324094081626637368299352404, 0.38324094081626637368299352404, 1.42597084608011824219471436529, 1.67262753856399767295504209267, 1.77508943274160610130662980499, 3.00090254452484019844807963947, 3.02839644430181302316841523805, 3.48574509724151396245479699417, 3.68338034009419020565675148433, 4.46710771200692897097441782903, 4.70918558880949836492788608586, 4.84661700716756271430388799358, 5.62028951728929534403938904447, 5.84007812889460138407102925505, 6.38280924394720604513236859033, 6.68841949709193800757686584592, 6.96476330116179416107196007825, 7.24758293111981625403311020988, 7.63729536031500328498681463323, 8.084053123728582976026940773758, 8.511440155065986906022627686112

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