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Properties

Label 4-800e2-1.1-c1e2-0-33
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  + 2·9-s + 8·11-s + 16·19-s + 4·29-s − 8·31-s − 20·41-s + 10·49-s + 4·61-s + 24·71-s − 16·79-s − 5·81-s + 12·89-s + 16·99-s + 28·101-s + 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 32·171-s + ⋯
L(s)  = 1  + 2/3·9-s + 2.41·11-s + 3.67·19-s + 0.742·29-s − 1.43·31-s − 3.12·41-s + 10/7·49-s + 0.512·61-s + 2.84·71-s − 1.80·79-s − 5/9·81-s + 1.27·89-s + 1.60·99-s + 2.78·101-s + 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 2.44·171-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

\(L(1)\) \(\approx\) \(2.972109543\)
\(L(\frac12)\) \(\approx\) \(2.972109543\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.19.aq_dy
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.43.a_ade
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \) 2.47.a_adm
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.67.a_adu
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.83.a_aco
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−10.07504478878799993589527893617, −10.07494148824040044632683123428, −9.754399181016294135641612742267, −9.147593313744008898440551813804, −8.879208929021368555566973043790, −8.589731752532433791165519710500, −7.69092572792073251787856259954, −7.44420438790112111466250655457, −7.02180447652963304443323578739, −6.71305447448958984361620162138, −6.15556154248036396710764626554, −5.61394579599834384622783529656, −4.96530917422889623830417257180, −4.86898791580292905415471260025, −3.74305977784943563899566738819, −3.67468791370936355590713059598, −3.28131717218369490551418823091, −2.21195462598258886970323004006, −1.30114715274294811059575399550, −1.11332336049578327972894929450, 1.11332336049578327972894929450, 1.30114715274294811059575399550, 2.21195462598258886970323004006, 3.28131717218369490551418823091, 3.67468791370936355590713059598, 3.74305977784943563899566738819, 4.86898791580292905415471260025, 4.96530917422889623830417257180, 5.61394579599834384622783529656, 6.15556154248036396710764626554, 6.71305447448958984361620162138, 7.02180447652963304443323578739, 7.44420438790112111466250655457, 7.69092572792073251787856259954, 8.589731752532433791165519710500, 8.879208929021368555566973043790, 9.147593313744008898440551813804, 9.754399181016294135641612742267, 10.07494148824040044632683123428, 10.07504478878799993589527893617

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