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Properties

Label 4-252e2-1.1-c7e2-0-3
Degree 44
Conductor 6350463504
Sign 11
Analytic cond. 6197.006197.00
Root an. cond. 8.872488.87248
Motivic weight 77
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 264·5-s + 686·7-s + 4.98e3·11-s − 1.01e4·13-s − 1.78e4·17-s + 6.25e3·19-s − 1.40e4·23-s + 2.73e4·25-s − 2.43e5·29-s + 4.70e5·31-s − 1.81e5·35-s + 3.11e5·37-s − 9.19e5·41-s − 1.12e5·43-s + 1.02e5·47-s + 3.52e5·49-s − 2.72e6·53-s − 1.31e6·55-s + 1.90e4·59-s − 9.25e5·61-s + 2.67e6·65-s − 2.05e6·67-s + 8.69e5·71-s + 3.50e6·73-s + 3.41e6·77-s − 6.64e6·79-s − 7.85e6·83-s + ⋯
L(s)  = 1  − 0.944·5-s + 0.755·7-s + 1.12·11-s − 1.28·13-s − 0.880·17-s + 0.209·19-s − 0.240·23-s + 0.350·25-s − 1.85·29-s + 2.83·31-s − 0.713·35-s + 1.00·37-s − 2.08·41-s − 0.216·43-s + 0.143·47-s + 3/7·49-s − 2.50·53-s − 1.06·55-s + 0.0120·59-s − 0.521·61-s + 1.21·65-s − 0.834·67-s + 0.288·71-s + 1.05·73-s + 0.852·77-s − 1.51·79-s − 1.50·83-s + ⋯

Functional equation

Λ(s)=(63504s/2 ΓC(s)2 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(63504s/2 ΓC(s+7/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 6350463504    =    24⋅34⋅722^{4} \cdot 3^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 6197.006197.00
Root analytic conductor: 8.872488.87248
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 63504, ( :7/2,7/2), 1)(4,\ 63504,\ (\ :7/2, 7/2),\ 1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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
7C1C_1 (1−p3T)2 ( 1 - p^{3} T )^{2}
good5D4D_{4} 1+264T+8462pT2+264p7T3+p14T4 1 + 264 T + 8462 p T^{2} + 264 p^{7} T^{3} + p^{14} T^{4}
11D4D_{4} 1−4980T+38737606T2−4980p7T3+p14T4 1 - 4980 T + 38737606 T^{2} - 4980 p^{7} T^{3} + p^{14} T^{4}
13D4D_{4} 1+10148T+75576846T2+10148p7T3+p14T4 1 + 10148 T + 75576846 T^{2} + 10148 p^{7} T^{3} + p^{14} T^{4}
17D4D_{4} 1+17832T+345159502T2+17832p7T3+p14T4 1 + 17832 T + 345159502 T^{2} + 17832 p^{7} T^{3} + p^{14} T^{4}
19D4D_{4} 1−6256T+1754965926T2−6256p7T3+p14T4 1 - 6256 T + 1754965926 T^{2} - 6256 p^{7} T^{3} + p^{14} T^{4}
23D4D_{4} 1+14052T+6233591566T2+14052p7T3+p14T4 1 + 14052 T + 6233591566 T^{2} + 14052 p^{7} T^{3} + p^{14} T^{4}
29D4D_{4} 1+243588T+49005121054T2+243588p7T3+p14T4 1 + 243588 T + 49005121054 T^{2} + 243588 p^{7} T^{3} + p^{14} T^{4}
31D4D_{4} 1−470824T+110401476030T2−470824p7T3+p14T4 1 - 470824 T + 110401476030 T^{2} - 470824 p^{7} T^{3} + p^{14} T^{4}
37D4D_{4} 1−311116T+134140188030T2−311116p7T3+p14T4 1 - 311116 T + 134140188030 T^{2} - 311116 p^{7} T^{3} + p^{14} T^{4}
41D4D_{4} 1+919248T+579002453902T2+919248p7T3+p14T4 1 + 919248 T + 579002453902 T^{2} + 919248 p^{7} T^{3} + p^{14} T^{4}
43D4D_{4} 1+112616T+296638211478T2+112616p7T3+p14T4 1 + 112616 T + 296638211478 T^{2} + 112616 p^{7} T^{3} + p^{14} T^{4}
47D4D_{4} 1−102456T+463813338910T2−102456p7T3+p14T4 1 - 102456 T + 463813338910 T^{2} - 102456 p^{7} T^{3} + p^{14} T^{4}
53D4D_{4} 1+2720028T+3726830950846T2+2720028p7T3+p14T4 1 + 2720028 T + 3726830950846 T^{2} + 2720028 p^{7} T^{3} + p^{14} T^{4}
59D4D_{4} 1−19008T−1663332768746T2−19008p7T3+p14T4 1 - 19008 T - 1663332768746 T^{2} - 19008 p^{7} T^{3} + p^{14} T^{4}
61D4D_{4} 1+925148T+2505443574174T2+925148p7T3+p14T4 1 + 925148 T + 2505443574174 T^{2} + 925148 p^{7} T^{3} + p^{14} T^{4}
67D4D_{4} 1+2053424T+3053533142214T2+2053424p7T3+p14T4 1 + 2053424 T + 3053533142214 T^{2} + 2053424 p^{7} T^{3} + p^{14} T^{4}
71D4D_{4} 1−869508T+14567098422382T2−869508p7T3+p14T4 1 - 869508 T + 14567098422382 T^{2} - 869508 p^{7} T^{3} + p^{14} T^{4}
73D4D_{4} 1−3505228T+11289516695814T2−3505228p7T3+p14T4 1 - 3505228 T + 11289516695814 T^{2} - 3505228 p^{7} T^{3} + p^{14} T^{4}
79D4D_{4} 1+6640856T+37612376152158T2+6640856p7T3+p14T4 1 + 6640856 T + 37612376152158 T^{2} + 6640856 p^{7} T^{3} + p^{14} T^{4}
83D4D_{4} 1+7856760T+50010766932550T2+7856760p7T3+p14T4 1 + 7856760 T + 50010766932550 T^{2} + 7856760 p^{7} T^{3} + p^{14} T^{4}
89D4D_{4} 1−9330384T+109971781045486T2−9330384p7T3+p14T4 1 - 9330384 T + 109971781045486 T^{2} - 9330384 p^{7} T^{3} + p^{14} T^{4}
97D4D_{4} 1+2220212T+72228119801046T2+2220212p7T3+p14T4 1 + 2220212 T + 72228119801046 T^{2} + 2220212 p^{7} T^{3} + p^{14} 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

−10.57526851572705075479432081697, −10.09044807573988639224395868718, −9.497938331525630254815875111195, −9.202171084692448943749368721434, −8.495096268585817997387706021945, −8.071878988251383391394120417095, −7.67721206045429886701212209214, −7.19742349634444734554011499595, −6.42100940558754194034163469971, −6.35661771896507738405290366689, −5.14765097337015763652225361744, −4.93954190247693560846597331913, −4.17244927374621037037280666814, −3.99813506580270940881136839651, −3.03444279862126322902583779027, −2.51020183650228963671319030700, −1.61919959239364366267299229392, −1.19770680018989848264435622719, 0, 0, 1.19770680018989848264435622719, 1.61919959239364366267299229392, 2.51020183650228963671319030700, 3.03444279862126322902583779027, 3.99813506580270940881136839651, 4.17244927374621037037280666814, 4.93954190247693560846597331913, 5.14765097337015763652225361744, 6.35661771896507738405290366689, 6.42100940558754194034163469971, 7.19742349634444734554011499595, 7.67721206045429886701212209214, 8.071878988251383391394120417095, 8.495096268585817997387706021945, 9.202171084692448943749368721434, 9.497938331525630254815875111195, 10.09044807573988639224395868718, 10.57526851572705075479432081697

Graph of the ZZ-function along the critical line