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Properties

Label 4-800e2-1.1-c1e2-0-32
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  + 4·7-s + 5·9-s + 6·17-s − 8·23-s + 16·31-s + 14·41-s + 4·47-s − 2·49-s + 20·63-s − 4·71-s − 2·73-s − 20·79-s + 16·81-s + 10·89-s − 4·97-s − 28·103-s + 18·113-s + 24·119-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 30·153-s + 157-s + ⋯
L(s)  = 1  + 1.51·7-s + 5/3·9-s + 1.45·17-s − 1.66·23-s + 2.87·31-s + 2.18·41-s + 0.583·47-s − 2/7·49-s + 2.51·63-s − 0.474·71-s − 0.234·73-s − 2.25·79-s + 16/9·81-s + 1.05·89-s − 0.406·97-s − 2.75·103-s + 1.69·113-s + 2.20·119-s − 0.272·121-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 + 2.42·153-s + 0.0798·157-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

\(L(1)\) \(\approx\) \(3.221897512\)
\(L(\frac12)\) \(\approx\) \(3.221897512\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.11.a_d
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.19.a_abl
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.41.ao_fb
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.47.ae_du
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.a_adm
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.59.a_ady
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \) 2.67.a_aev
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.71.e_fq
73$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.73.c_fr
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.83.a_adh
89$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.89.ak_hv
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−10.12872586785511455819898242038, −10.12649932084787843853032669339, −9.917304737506504149741029047131, −9.297665563600014284019663955692, −8.664475628918506523795359450115, −8.214773337624995245286678568277, −7.77794689921965347347444367992, −7.73174710660467376350285325656, −7.16598641907526934801663454343, −6.58943013681872460097654020463, −5.98182785417239352563257432268, −5.75691572121847084707051741112, −4.87206412073795270330000754109, −4.67137761843984415728019177648, −4.18530548021858398147975876455, −3.78911440117897465656878749884, −2.86131984084813309866341840856, −2.25040969906291522765180708379, −1.40368579491327865028620211999, −1.10695232162380115181015105749, 1.10695232162380115181015105749, 1.40368579491327865028620211999, 2.25040969906291522765180708379, 2.86131984084813309866341840856, 3.78911440117897465656878749884, 4.18530548021858398147975876455, 4.67137761843984415728019177648, 4.87206412073795270330000754109, 5.75691572121847084707051741112, 5.98182785417239352563257432268, 6.58943013681872460097654020463, 7.16598641907526934801663454343, 7.73174710660467376350285325656, 7.77794689921965347347444367992, 8.214773337624995245286678568277, 8.664475628918506523795359450115, 9.297665563600014284019663955692, 9.917304737506504149741029047131, 10.12649932084787843853032669339, 10.12872586785511455819898242038

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