Properties

Label 4-40e4-1.1-c1e2-0-38
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $163.227$
Root an. cond. $3.57436$
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  + 2·3-s + 4·7-s + 2·9-s − 2·11-s − 2·13-s + 6·19-s + 8·21-s + 12·23-s + 6·27-s − 6·29-s + 16·31-s − 4·33-s + 6·37-s − 4·39-s + 10·43-s − 2·49-s + 10·53-s + 12·57-s − 6·59-s − 18·61-s + 8·63-s − 10·67-s + 24·69-s + 8·73-s − 8·77-s + 11·81-s + 2·83-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.51·7-s + 2/3·9-s − 0.603·11-s − 0.554·13-s + 1.37·19-s + 1.74·21-s + 2.50·23-s + 1.15·27-s − 1.11·29-s + 2.87·31-s − 0.696·33-s + 0.986·37-s − 0.640·39-s + 1.52·43-s − 2/7·49-s + 1.37·53-s + 1.58·57-s − 0.781·59-s − 2.30·61-s + 1.00·63-s − 1.22·67-s + 2.88·69-s + 0.936·73-s − 0.911·77-s + 11/9·81-s + 0.219·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(163.227\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.368365887\)
\(L(\frac12)\) \(\approx\) \(5.368365887\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
11$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_c
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.c_c
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_s
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.g_s
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_s
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_by
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.ak_by
59$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_s
61$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.61.s_gg
67$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_by
71$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.71.a_abq
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.83.ac_c
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \) 2.89.a_agg
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

−9.297236129656082950219034357114, −9.271574097510806860633261607808, −8.761414318824999894348447805031, −8.460311740436829434519280396047, −7.934266578093900937375458335451, −7.70344864331412648208238040484, −7.34469432582924521459797946217, −7.17609075176525688818437269853, −6.25569424794157170047928943777, −6.11345292999740449245689439992, −5.11871327533801936140181988879, −5.11755661289014142830855737583, −4.55024001762475265967683469123, −4.41800258016212945509248578004, −3.35078832512169742578649991236, −3.13812131818758889137264684957, −2.57684714268305765122376662778, −2.27531674827328281987956498482, −1.23374554061319535138905960430, −1.02201724135173927526339978237, 1.02201724135173927526339978237, 1.23374554061319535138905960430, 2.27531674827328281987956498482, 2.57684714268305765122376662778, 3.13812131818758889137264684957, 3.35078832512169742578649991236, 4.41800258016212945509248578004, 4.55024001762475265967683469123, 5.11755661289014142830855737583, 5.11871327533801936140181988879, 6.11345292999740449245689439992, 6.25569424794157170047928943777, 7.17609075176525688818437269853, 7.34469432582924521459797946217, 7.70344864331412648208238040484, 7.934266578093900937375458335451, 8.460311740436829434519280396047, 8.761414318824999894348447805031, 9.271574097510806860633261607808, 9.297236129656082950219034357114

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