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Properties

Label 4-1800e2-1.1-c3e2-0-31
Degree 44
Conductor 32400003240000
Sign 11
Analytic cond. 11279.111279.1
Root an. cond. 10.305510.3055
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·7-s − 8·11-s + 14·13-s + 86·19-s − 128·23-s − 344·29-s − 158·31-s + 36·37-s + 244·41-s + 390·43-s − 756·47-s + 65·49-s − 268·53-s − 4·59-s − 1.03e3·61-s + 1.62e3·67-s + 276·71-s + 644·73-s − 48·77-s − 944·79-s + 484·83-s − 2.22e3·89-s + 84·91-s + 510·97-s + 844·101-s + 544·103-s − 3.22e3·107-s + ⋯
L(s)  = 1  + 0.323·7-s − 0.219·11-s + 0.298·13-s + 1.03·19-s − 1.16·23-s − 2.20·29-s − 0.915·31-s + 0.159·37-s + 0.929·41-s + 1.38·43-s − 2.34·47-s + 0.189·49-s − 0.694·53-s − 0.00882·59-s − 2.17·61-s + 2.95·67-s + 0.461·71-s + 1.03·73-s − 0.0710·77-s − 1.34·79-s + 0.640·83-s − 2.64·89-s + 0.0967·91-s + 0.533·97-s + 0.831·101-s + 0.520·103-s − 2.90·107-s + ⋯

Functional equation

Λ(s)=(3240000s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(3240000s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 32400003240000    =    26⋅34⋅542^{6} \cdot 3^{4} \cdot 5^{4}
Sign: 11
Analytic conductor: 11279.111279.1
Root analytic conductor: 10.305510.3055
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 3240000, ( :3/2,3/2), 1)(4,\ 3240000,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
5 1 1
good7D4D_{4} 1−6T−29T2−6p3T3+p6T4 1 - 6 T - 29 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+8T+1954T2+8p3T3+p6T4 1 + 8 T + 1954 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−14T+119pT2−14p3T3+p6T4 1 - 14 T + 119 p T^{2} - 14 p^{3} T^{3} + p^{6} T^{4}
17C22C_2^2 1+9102T2+p6T4 1 + 9102 T^{2} + p^{6} T^{4}
19D4D_{4} 1−86T+9051T2−86p3T3+p6T4 1 - 86 T + 9051 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+128T+21914T2+128p3T3+p6T4 1 + 128 T + 21914 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+344T+2078pT2+344p3T3+p6T4 1 + 344 T + 2078 p T^{2} + 344 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+158T+30347T2+158p3T3+p6T4 1 + 158 T + 30347 T^{2} + 158 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−36T+75566T2−36p3T3+p6T4 1 - 36 T + 75566 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−244T+117250T2−244p3T3+p6T4 1 - 244 T + 117250 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−390T+161563T2−390p3T3+p6T4 1 - 390 T + 161563 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+756T+338946T2+756p3T3+p6T4 1 + 756 T + 338946 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+268T+280234T2+268p3T3+p6T4 1 + 268 T + 280234 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+4T+364426T2+4p3T3+p6T4 1 + 4 T + 364426 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+1034T+674915T2+1034p3T3+p6T4 1 + 1034 T + 674915 T^{2} + 1034 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−1622T+1200603T2−1622p3T3+p6T4 1 - 1622 T + 1200603 T^{2} - 1622 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−276T−53570T2−276p3T3+p6T4 1 - 276 T - 53570 T^{2} - 276 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−644T+809318T2−644p3T3+p6T4 1 - 644 T + 809318 T^{2} - 644 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+944T+919262T2+944p3T3+p6T4 1 + 944 T + 919262 T^{2} + 944 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−484T+460762T2−484p3T3+p6T4 1 - 484 T + 460762 T^{2} - 484 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+2224T+2634898T2+2224p3T3+p6T4 1 + 2224 T + 2634898 T^{2} + 2224 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−510T+1148995T2−510p3T3+p6T4 1 - 510 T + 1148995 T^{2} - 510 p^{3} T^{3} + p^{6} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(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

−8.659200095688868914838211273992, −8.327845637601000965541646538534, −7.74596938661892142430171237271, −7.63271301677198280093620835408, −7.36845267733074308784211305340, −6.65650270458210145966859831172, −6.32072133532544498638575214906, −5.85651619189362015616890196507, −5.43198688167205370744747466064, −5.18899958515447941859443083447, −4.60974227012599161546700276652, −4.06942302483626015670741981293, −3.59817969884900890684348407779, −3.41763886745951370487965166031, −2.48761835565738179474531515071, −2.26509670378497213469723037291, −1.43356289792186708397806462179, −1.21511692913021293780627915718, 0, 0, 1.21511692913021293780627915718, 1.43356289792186708397806462179, 2.26509670378497213469723037291, 2.48761835565738179474531515071, 3.41763886745951370487965166031, 3.59817969884900890684348407779, 4.06942302483626015670741981293, 4.60974227012599161546700276652, 5.18899958515447941859443083447, 5.43198688167205370744747466064, 5.85651619189362015616890196507, 6.32072133532544498638575214906, 6.65650270458210145966859831172, 7.36845267733074308784211305340, 7.63271301677198280093620835408, 7.74596938661892142430171237271, 8.327845637601000965541646538534, 8.659200095688868914838211273992

Graph of the ZZ-function along the critical line