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Properties

Label 4-720000-1.1-c1e2-0-5
Degree $4$
Conductor $720000$
Sign $1$
Analytic cond. $45.9078$
Root an. cond. $2.60298$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s − 13-s − 4·17-s − 13·29-s + 13·37-s − 2·41-s + 11·49-s − 6·53-s + 61-s + 81-s + 3·89-s + 16·97-s − 101-s + 16·109-s − 6·113-s − 117-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1/3·9-s − 0.277·13-s − 0.970·17-s − 2.41·29-s + 2.13·37-s − 0.312·41-s + 11/7·49-s − 0.824·53-s + 0.128·61-s + 1/9·81-s + 0.317·89-s + 1.62·97-s − 0.0995·101-s + 1.53·109-s − 0.564·113-s − 0.0924·117-s + 4/11·121-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 − 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(720000\)    =    \(2^{7} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(45.9078\)
Root analytic conductor: \(2.60298\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 720000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.583374277\)
\(L(\frac12)\) \(\approx\) \(1.583374277\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.a_al
11$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.11.a_ae
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.13.b_u
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.e_n
19$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.19.a_ai
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.n_dq
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.a_n
37$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.37.an_eg
41$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.c_cg
43$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.43.a_be
47$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.47.a_t
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.53.g_dm
59$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.59.a_ai
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.61.ab_co
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2^2$ \( 1 - 137 T^{2} + p^{2} T^{4} \) 2.71.a_afh
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2^2$ \( 1 + 69 T^{2} + p^{2} T^{4} \) 2.79.a_cr
83$C_2^2$ \( 1 - 60 T^{2} + p^{2} T^{4} \) 2.83.a_aci
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.89.ad_y
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.aq_gc
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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

−8.304126583266294621780630197063, −7.74570056641618047210418698125, −7.37984147271246712941288426266, −7.16315557864195614633489541545, −6.36716339060585897194511547465, −6.17541782640688396932501617759, −5.55871019129010756089718522046, −5.11949964437806480749941852055, −4.47423252236251737624200414178, −4.13034430709533532271265823309, −3.59914762773414848340718450882, −2.88923088454388395311825499669, −2.22513293712655287939883331698, −1.73499233763016733927354991889, −0.61123201480997862739229051511, 0.61123201480997862739229051511, 1.73499233763016733927354991889, 2.22513293712655287939883331698, 2.88923088454388395311825499669, 3.59914762773414848340718450882, 4.13034430709533532271265823309, 4.47423252236251737624200414178, 5.11949964437806480749941852055, 5.55871019129010756089718522046, 6.17541782640688396932501617759, 6.36716339060585897194511547465, 7.16315557864195614633489541545, 7.37984147271246712941288426266, 7.74570056641618047210418698125, 8.304126583266294621780630197063

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