Properties

Label 4-336e2-1.1-c5e2-0-12
Degree 44
Conductor 112896112896
Sign 11
Analytic cond. 2904.022904.02
Root an. cond. 7.340917.34091
Motivic weight 55
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  + 18·3-s − 78·5-s + 98·7-s + 243·9-s + 370·11-s − 1.05e3·13-s − 1.40e3·15-s − 1.02e3·17-s − 280·19-s + 1.76e3·21-s − 1.93e3·23-s + 3.13e3·25-s + 2.91e3·27-s + 2.55e3·29-s − 9.40e3·31-s + 6.66e3·33-s − 7.64e3·35-s − 168·37-s − 1.90e4·39-s − 8.16e3·41-s − 2.65e4·43-s − 1.89e4·45-s − 3.33e4·47-s + 7.20e3·49-s − 1.84e4·51-s + 3.59e4·53-s − 2.88e4·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.39·5-s + 0.755·7-s + 9-s + 0.921·11-s − 1.73·13-s − 1.61·15-s − 0.861·17-s − 0.177·19-s + 0.872·21-s − 0.760·23-s + 1.00·25-s + 0.769·27-s + 0.564·29-s − 1.75·31-s + 1.06·33-s − 1.05·35-s − 0.0201·37-s − 2.00·39-s − 0.758·41-s − 2.18·43-s − 1.39·45-s − 2.20·47-s + 3/7·49-s − 0.994·51-s + 1.75·53-s − 1.28·55-s + ⋯

Functional equation

Λ(s)=(112896s/2ΓC(s)2L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(112896s/2ΓC(s+5/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 112896112896    =    2832722^{8} \cdot 3^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 2904.022904.02
Root analytic conductor: 7.340917.34091
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 112896, ( :5/2,5/2), 1)(4,\ 112896,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)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 (1p2T)2 ( 1 - p^{2} T )^{2}
7C1C_1 (1p2T)2 ( 1 - p^{2} T )^{2}
good5D4D_{4} 1+78T+2946T2+78p5T3+p10T4 1 + 78 T + 2946 T^{2} + 78 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 1370T+92110T2370p5T3+p10T4 1 - 370 T + 92110 T^{2} - 370 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1+1056T+1002070T2+1056p5T3+p10T4 1 + 1056 T + 1002070 T^{2} + 1056 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 1+1026T+3101146T2+1026p5T3+p10T4 1 + 1026 T + 3101146 T^{2} + 1026 p^{5} T^{3} + p^{10} T^{4}
19D4D_{4} 1+280T+4527126T2+280p5T3+p10T4 1 + 280 T + 4527126 T^{2} + 280 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 1+1930T+8676094T2+1930p5T3+p10T4 1 + 1930 T + 8676094 T^{2} + 1930 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 12556T+42210910T22556p5T3+p10T4 1 - 2556 T + 42210910 T^{2} - 2556 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 1+9408T+74445118T2+9408p5T3+p10T4 1 + 9408 T + 74445118 T^{2} + 9408 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 1+168T25371242T2+168p5T3+p10T4 1 + 168 T - 25371242 T^{2} + 168 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 1+8166T+110576466T2+8166p5T3+p10T4 1 + 8166 T + 110576466 T^{2} + 8166 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1+26552T+467848070T2+26552p5T3+p10T4 1 + 26552 T + 467848070 T^{2} + 26552 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1+33324T+728437086T2+33324p5T3+p10T4 1 + 33324 T + 728437086 T^{2} + 33324 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 135904T+1106075878T235904p5T3+p10T4 1 - 35904 T + 1106075878 T^{2} - 35904 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1332pT+633628534T2332p6T3+p10T4 1 - 332 p T + 633628534 T^{2} - 332 p^{6} T^{3} + p^{10} T^{4}
61D4D_{4} 1+39028T+2066826686T2+39028p5T3+p10T4 1 + 39028 T + 2066826686 T^{2} + 39028 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 1+57692T+3462397958T2+57692p5T3+p10T4 1 + 57692 T + 3462397958 T^{2} + 57692 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1+9558T+1530801150T2+9558p5T3+p10T4 1 + 9558 T + 1530801150 T^{2} + 9558 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 18712T1359024578T28712p5T3+p10T4 1 - 8712 T - 1359024578 T^{2} - 8712 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 1+33420T+6193824670T2+33420p5T3+p10T4 1 + 33420 T + 6193824670 T^{2} + 33420 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 1+17752T+7945747862T2+17752p5T3+p10T4 1 + 17752 T + 7945747862 T^{2} + 17752 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1+167718T+17824867954T2+167718p5T3+p10T4 1 + 167718 T + 17824867954 T^{2} + 167718 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1+46928T+163241630pT2+46928p5T3+p10T4 1 + 46928 T + 163241630 p T^{2} + 46928 p^{5} T^{3} + p^{10} T^{4}
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   L(s)=p j=14(1αj,pps)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

−10.31685558446154178310715702040, −10.05495717819009732396194948904, −9.483832325207236552175598675228, −8.924945292991519994465890686364, −8.519059911660735179091722664641, −8.244158889013104205174064241731, −7.62157680970586809830829996555, −7.31815335625968647044186042610, −6.87796205081820817524132961498, −6.36910601865660901372724013143, −5.22695060429923522237596925266, −4.89892099513590356478160263040, −4.18150844105286706236335997196, −3.98640605820107238500241488748, −3.24985450308796590422261284450, −2.65931988772819559974016718986, −1.85106190556346480537127101507, −1.45650547824126596360619039640, 0, 0, 1.45650547824126596360619039640, 1.85106190556346480537127101507, 2.65931988772819559974016718986, 3.24985450308796590422261284450, 3.98640605820107238500241488748, 4.18150844105286706236335997196, 4.89892099513590356478160263040, 5.22695060429923522237596925266, 6.36910601865660901372724013143, 6.87796205081820817524132961498, 7.31815335625968647044186042610, 7.62157680970586809830829996555, 8.244158889013104205174064241731, 8.519059911660735179091722664641, 8.924945292991519994465890686364, 9.483832325207236552175598675228, 10.05495717819009732396194948904, 10.31685558446154178310715702040

Graph of the ZZ-function along the critical line