Properties

Label 4-800e2-1.1-c1e2-0-12
Degree $4$
Conductor $640000$
Sign $1$
Analytic cond. $40.8069$
Root an. cond. $2.52745$
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  − 4·3-s + 8·9-s + 12·11-s + 8·17-s − 12·27-s − 48·33-s − 12·41-s − 12·43-s − 32·51-s + 12·67-s + 24·73-s + 23·81-s − 4·83-s + 24·97-s + 96·99-s − 28·107-s − 16·113-s + 86·121-s + 48·123-s + ⋯
L(s)  = 1  − 2.30·3-s + 8/3·9-s + 3.61·11-s + 1.94·17-s − 2.30·27-s − 8.35·33-s − 1.87·41-s − 1.82·43-s − 4.48·51-s + 1.46·67-s + 2.80·73-s + 23/9·81-s − 0.439·83-s + 2.43·97-s + 9.64·99-s − 2.70·107-s − 1.50·113-s + 7.81·121-s + 4.32·123-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(640000\)    =    \(2^{10} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(40.8069\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 640000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.135026936\)
\(L(\frac12)\) \(\approx\) \(1.135026936\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.3.e_i
7$C_2^2$ \( 1 + p^{2} T^{4} \) 2.7.a_a
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.11.am_cg
13$C_2^2$ \( 1 + p^{2} T^{4} \) 2.13.a_a
17$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bg
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.19.a_abi
23$C_2^2$ \( 1 + p^{2} T^{4} \) 2.23.a_a
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2^2$ \( 1 + p^{2} T^{4} \) 2.37.a_a
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_cu
47$C_2^2$ \( 1 + p^{2} T^{4} \) 2.47.a_a
53$C_2^2$ \( 1 + p^{2} T^{4} \) 2.53.a_a
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.59.a_ade
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.67.am_cu
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2^2$ \( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.73.ay_lc
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_i
89$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \) 2.89.a_fq
97$C_2^2$ \( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.97.ay_lc
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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

−10.48722080064354338540539401990, −10.09268022359926029866307532480, −9.791874234471254702659467099495, −9.201203727483320988190480055985, −9.138740062090640023502522260446, −8.150771758520677918582215325569, −8.054130074138071708319466951205, −7.01009412911764965826068222557, −6.74451280268357617460245481820, −6.65291589380958311300919344950, −6.07799681313080001687520879052, −5.73905525244797566231259606060, −5.16783668080927167766382205173, −4.89005584491219131574238981638, −4.14300174892925885753815429607, −3.53134546627626128718760516226, −3.51183439168114773091687953246, −1.82287663019988428222672353195, −1.31437944960221481870748300333, −0.73105651618821827495685605173, 0.73105651618821827495685605173, 1.31437944960221481870748300333, 1.82287663019988428222672353195, 3.51183439168114773091687953246, 3.53134546627626128718760516226, 4.14300174892925885753815429607, 4.89005584491219131574238981638, 5.16783668080927167766382205173, 5.73905525244797566231259606060, 6.07799681313080001687520879052, 6.65291589380958311300919344950, 6.74451280268357617460245481820, 7.01009412911764965826068222557, 8.054130074138071708319466951205, 8.150771758520677918582215325569, 9.138740062090640023502522260446, 9.201203727483320988190480055985, 9.791874234471254702659467099495, 10.09268022359926029866307532480, 10.48722080064354338540539401990

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