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Properties

Label 4-2900e2-1.1-c1e2-0-4
Degree $4$
Conductor $8410000$
Sign $1$
Analytic cond. $536.228$
Root an. cond. $4.81213$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·7-s + 2·9-s + 4·11-s − 4·13-s + 12·17-s − 8·19-s + 10·23-s − 4·29-s − 12·31-s + 14·41-s + 18·49-s + 8·53-s + 18·61-s − 12·63-s − 6·67-s − 28·73-s − 24·77-s − 16·79-s − 5·81-s + 6·83-s + 14·89-s + 24·91-s + 8·99-s − 18·101-s − 18·103-s − 10·107-s − 36·109-s + ⋯
L(s)  = 1  − 2.26·7-s + 2/3·9-s + 1.20·11-s − 1.10·13-s + 2.91·17-s − 1.83·19-s + 2.08·23-s − 0.742·29-s − 2.15·31-s + 2.18·41-s + 18/7·49-s + 1.09·53-s + 2.30·61-s − 1.51·63-s − 0.733·67-s − 3.27·73-s − 2.73·77-s − 1.80·79-s − 5/9·81-s + 0.658·83-s + 1.48·89-s + 2.51·91-s + 0.804·99-s − 1.79·101-s − 1.77·103-s − 0.966·107-s − 3.44·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8410000\)    =    \(2^{4} \cdot 5^{4} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(536.228\)
Root analytic conductor: \(4.81213\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8410000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.569185322\)
\(L(\frac12)\) \(\approx\) \(1.569185322\)
\(L(\frac{3}{2})\) 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 \( 1 \)
29$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.7.g_s
11$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_i
13$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_i
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.19.i_bg
23$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.23.ak_by
31$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_cu
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.37.a_abm
41$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.41.ao_du
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_bg
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.59.a_ady
61$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.61.as_gg
67$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_s
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.73.bc_ne
79$C_2^2$ \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.79.q_ey
83$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_s
89$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.89.ao_du
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−9.379768314640414115649867775725, −8.880613867348480001755859510985, −8.193529009059072443594106182959, −7.58515957344434026104251885667, −7.37921140321126875248277844210, −6.96433189869470152212222528860, −6.86538486156140705656128655942, −6.39807696666865248092829321780, −5.76657296274509949119021804799, −5.54661094904062123309254949361, −5.43582126961429306085295709244, −4.34333822966742514393820838722, −4.27967073720307300271273431835, −3.70798527963776588288731338273, −3.38524268383881273635851517533, −2.90102209465193200658671914853, −2.58166436293935809995596778845, −1.71678062882330372850353226323, −1.14984765122851659209047568091, −0.44204649216281090613912552256, 0.44204649216281090613912552256, 1.14984765122851659209047568091, 1.71678062882330372850353226323, 2.58166436293935809995596778845, 2.90102209465193200658671914853, 3.38524268383881273635851517533, 3.70798527963776588288731338273, 4.27967073720307300271273431835, 4.34333822966742514393820838722, 5.43582126961429306085295709244, 5.54661094904062123309254949361, 5.76657296274509949119021804799, 6.39807696666865248092829321780, 6.86538486156140705656128655942, 6.96433189869470152212222528860, 7.37921140321126875248277844210, 7.58515957344434026104251885667, 8.193529009059072443594106182959, 8.880613867348480001755859510985, 9.379768314640414115649867775725

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