Properties

Label 4-700e2-1.1-c1e2-0-4
Degree $4$
Conductor $490000$
Sign $1$
Analytic cond. $31.2428$
Root an. cond. $2.36421$
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  − 3·3-s − 7-s + 3·9-s + 2·11-s + 12·13-s + 2·17-s + 3·21-s − 9·23-s + 6·29-s − 2·31-s − 6·33-s + 8·37-s − 36·39-s + 10·41-s − 2·43-s + 8·47-s − 6·49-s − 6·51-s + 4·53-s + 8·59-s − 7·61-s − 3·63-s − 3·67-s + 27·69-s + 16·71-s + 14·73-s − 2·77-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.377·7-s + 9-s + 0.603·11-s + 3.32·13-s + 0.485·17-s + 0.654·21-s − 1.87·23-s + 1.11·29-s − 0.359·31-s − 1.04·33-s + 1.31·37-s − 5.76·39-s + 1.56·41-s − 0.304·43-s + 1.16·47-s − 6/7·49-s − 0.840·51-s + 0.549·53-s + 1.04·59-s − 0.896·61-s − 0.377·63-s − 0.366·67-s + 3.25·69-s + 1.89·71-s + 1.63·73-s − 0.227·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(490000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.2428\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 490000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.156778421\)
\(L(\frac12)\) \(\approx\) \(1.156778421\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) 2.3.d_g
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_ah
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_an
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.19.a_at
23$C_2^2$ \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.23.j_cg
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.29.ag_cp
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_abb
37$C_2^2$ \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_bb
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.41.ak_ed
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_r
53$C_2^2$ \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_abl
59$C_2^2$ \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_f
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_am
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.67.d_acg
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.73.ao_et
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.79.e_acl
83$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.83.ac_gl
89$C_2^2$ \( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.89.n_dc
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−10.73933745385851493502417177774, −10.54802795130702890349441450560, −9.862395671494668382720977579197, −9.486373957961796894308117388180, −8.945986645573651820304592625124, −8.501938853220002993874118682912, −8.063316455681579894919765870180, −7.72564588396874043121645627425, −6.79159175558741715181563945776, −6.48513338097050374821343449940, −6.06971771051922446566618561723, −5.88680034764386969265989558344, −5.60967500654854610223777264096, −4.80540911759818867226455012828, −4.06500472574838242289754385963, −3.86366605373192957546620990107, −3.29358468842960578616381354387, −2.26010940022169707386274187625, −1.23548129594604909976814531111, −0.75743476891547149038775290396, 0.75743476891547149038775290396, 1.23548129594604909976814531111, 2.26010940022169707386274187625, 3.29358468842960578616381354387, 3.86366605373192957546620990107, 4.06500472574838242289754385963, 4.80540911759818867226455012828, 5.60967500654854610223777264096, 5.88680034764386969265989558344, 6.06971771051922446566618561723, 6.48513338097050374821343449940, 6.79159175558741715181563945776, 7.72564588396874043121645627425, 8.063316455681579894919765870180, 8.501938853220002993874118682912, 8.945986645573651820304592625124, 9.486373957961796894308117388180, 9.862395671494668382720977579197, 10.54802795130702890349441450560, 10.73933745385851493502417177774

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