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Properties

Label 4-600e2-1.1-c3e2-0-10
Degree 44
Conductor 360000360000
Sign 11
Analytic cond. 1253.241253.24
Root an. cond. 5.949885.94988
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 2·7-s + 27·9-s + 74·11-s + 98·13-s − 78·17-s − 80·19-s − 12·21-s + 40·23-s + 108·27-s + 50·29-s − 12·31-s + 444·33-s − 34·37-s + 588·39-s + 344·41-s + 216·43-s + 876·47-s + 478·49-s − 468·51-s − 634·53-s − 480·57-s + 666·59-s + 244·61-s − 54·63-s − 980·67-s + 240·69-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.107·7-s + 9-s + 2.02·11-s + 2.09·13-s − 1.11·17-s − 0.965·19-s − 0.124·21-s + 0.362·23-s + 0.769·27-s + 0.320·29-s − 0.0695·31-s + 2.34·33-s − 0.151·37-s + 2.41·39-s + 1.31·41-s + 0.766·43-s + 2.71·47-s + 1.39·49-s − 1.28·51-s − 1.64·53-s − 1.11·57-s + 1.46·59-s + 0.512·61-s − 0.107·63-s − 1.78·67-s + 0.418·69-s + ⋯

Functional equation

Λ(s)=(360000s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(360000s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 360000360000    =    26⋅32⋅542^{6} \cdot 3^{2} \cdot 5^{4}
Sign: 11
Analytic conductor: 1253.241253.24
Root analytic conductor: 5.949885.94988
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 360000, ( :3/2,3/2), 1)(4,\ 360000,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 7.0320685077.032068507
L(12)L(\frac12) ≈\approx 7.0320685077.032068507
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
3C1C_1 (1−pT)2 ( 1 - p T )^{2}
5 1 1
good7D4D_{4} 1+2T−474T2+2p3T3+p6T4 1 + 2 T - 474 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−74T+3902T2−74p3T3+p6T4 1 - 74 T + 3902 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−98T+6666T2−98p3T3+p6T4 1 - 98 T + 6666 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+78T+8122T2+78p3T3+p6T4 1 + 78 T + 8122 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+80T+7062T2+80p3T3+p6T4 1 + 80 T + 7062 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−40T+16478T2−40p3T3+p6T4 1 - 40 T + 16478 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−50T+43082T2−50p3T3+p6T4 1 - 50 T + 43082 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+12T+17822T2+12p3T3+p6T4 1 + 12 T + 17822 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+34T+20970T2+34p3T3+p6T4 1 + 34 T + 20970 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−344T+162782T2−344p3T3+p6T4 1 - 344 T + 162782 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4}
43C2C_2 (1−108T+p3T2)2 ( 1 - 108 T + p^{3} T^{2} )^{2}
47D4D_{4} 1−876T+374206T2−876p3T3+p6T4 1 - 876 T + 374206 T^{2} - 876 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+634T+398114T2+634p3T3+p6T4 1 + 634 T + 398114 T^{2} + 634 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−666T+211918T2−666p3T3+p6T4 1 - 666 T + 211918 T^{2} - 666 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−4pT+336750T2−4p4T3+p6T4 1 - 4 p T + 336750 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4}
67D4D_{4} 1+980T+692502T2+980p3T3+p6T4 1 + 980 T + 692502 T^{2} + 980 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−308T+553262T2−308p3T3+p6T4 1 - 308 T + 553262 T^{2} - 308 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+1412T+1090194T2+1412p3T3+p6T4 1 + 1412 T + 1090194 T^{2} + 1412 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−1052T+1237470T2−1052p3T3+p6T4 1 - 1052 T + 1237470 T^{2} - 1052 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−248T+1125926T2−248p3T3+p6T4 1 - 248 T + 1125926 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−684T+1229686T2−684p3T3+p6T4 1 - 684 T + 1229686 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+1840T+2539650T2+1840p3T3+p6T4 1 + 1840 T + 2539650 T^{2} + 1840 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

−10.43669562557646710216880811053, −10.09245799164554459136708975461, −9.144174632756916275628198108521, −9.007976491055808414617689994244, −8.945264403197844514153305746816, −8.604158444337866192502816749637, −7.82508521488061386966684652062, −7.49158388172521085927568310919, −6.73167191810326642606979893971, −6.59977515281625863186987996291, −6.02489169493280744427615631852, −5.69008575347177264158215849218, −4.39103178831439506798597622572, −4.36659344771878606735265885512, −3.79091446818388749661376483537, −3.45150280540685777266176179308, −2.57134343184453226264169563816, −2.05391789906745569314344577312, −1.25673155729579359690571827219, −0.827335399725095797662936038173, 0.827335399725095797662936038173, 1.25673155729579359690571827219, 2.05391789906745569314344577312, 2.57134343184453226264169563816, 3.45150280540685777266176179308, 3.79091446818388749661376483537, 4.36659344771878606735265885512, 4.39103178831439506798597622572, 5.69008575347177264158215849218, 6.02489169493280744427615631852, 6.59977515281625863186987996291, 6.73167191810326642606979893971, 7.49158388172521085927568310919, 7.82508521488061386966684652062, 8.604158444337866192502816749637, 8.945264403197844514153305746816, 9.007976491055808414617689994244, 9.144174632756916275628198108521, 10.09245799164554459136708975461, 10.43669562557646710216880811053

Graph of the ZZ-function along the critical line