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Properties

Label 4-70e2-1.1-c7e2-0-8
Degree 44
Conductor 49004900
Sign 11
Analytic cond. 478.163478.163
Root an. cond. 4.676214.67621
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  − 16·2-s + 45·3-s + 192·4-s + 250·5-s − 720·6-s − 686·7-s − 2.04e3·8-s − 2.50e3·9-s − 4.00e3·10-s − 2.84e3·11-s + 8.64e3·12-s + 753·13-s + 1.09e4·14-s + 1.12e4·15-s + 2.04e4·16-s − 6.41e3·17-s + 4.00e4·18-s + 2.50e3·19-s + 4.80e4·20-s − 3.08e4·21-s + 4.55e4·22-s − 3.11e4·23-s − 9.21e4·24-s + 4.68e4·25-s − 1.20e4·26-s − 2.18e5·27-s − 1.31e5·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.962·3-s + 3/2·4-s + 0.894·5-s − 1.36·6-s − 0.755·7-s − 1.41·8-s − 1.14·9-s − 1.26·10-s − 0.644·11-s + 1.44·12-s + 0.0950·13-s + 1.06·14-s + 0.860·15-s + 5/4·16-s − 0.316·17-s + 1.61·18-s + 0.0836·19-s + 1.34·20-s − 0.727·21-s + 0.911·22-s − 0.533·23-s − 1.36·24-s + 3/5·25-s − 0.134·26-s − 2.13·27-s − 1.13·28-s + ⋯

Functional equation

Λ(s)=(4900s/2 ΓC(s)2 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(4900s/2 ΓC(s+7/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 49004900    =    22⋅52⋅722^{2} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 478.163478.163
Root analytic conductor: 4.676214.67621
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 4900, ( :7/2,7/2), 1)(4,\ 4900,\ (\ :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)
bad2C1C_1 (1+p3T)2 ( 1 + p^{3} T )^{2}
5C1C_1 (1−p3T)2 ( 1 - p^{3} T )^{2}
7C1C_1 (1+p3T)2 ( 1 + p^{3} T )^{2}
good3D4D_{4} 1−5p2T+1510pT2−5p9T3+p14T4 1 - 5 p^{2} T + 1510 p T^{2} - 5 p^{9} T^{3} + p^{14} T^{4}
11D4D_{4} 1+2845T−6692542T2+2845p7T3+p14T4 1 + 2845 T - 6692542 T^{2} + 2845 p^{7} T^{3} + p^{14} T^{4}
13D4D_{4} 1−753T+77948396T2−753p7T3+p14T4 1 - 753 T + 77948396 T^{2} - 753 p^{7} T^{3} + p^{14} T^{4}
17D4D_{4} 1+6413T+58578932T2+6413p7T3+p14T4 1 + 6413 T + 58578932 T^{2} + 6413 p^{7} T^{3} + p^{14} T^{4}
19D4D_{4} 1−2502T+1748341838T2−2502p7T3+p14T4 1 - 2502 T + 1748341838 T^{2} - 2502 p^{7} T^{3} + p^{14} T^{4}
23D4D_{4} 1+31158T+461266174T2+31158p7T3+p14T4 1 + 31158 T + 461266174 T^{2} + 31158 p^{7} T^{3} + p^{14} T^{4}
29D4D_{4} 1+72699T+27821734708T2+72699p7T3+p14T4 1 + 72699 T + 27821734708 T^{2} + 72699 p^{7} T^{3} + p^{14} T^{4}
31D4D_{4} 1+10472pT+64287828414T2+10472p8T3+p14T4 1 + 10472 p T + 64287828414 T^{2} + 10472 p^{8} T^{3} + p^{14} T^{4}
37D4D_{4} 1+776108T+340404268782T2+776108p7T3+p14T4 1 + 776108 T + 340404268782 T^{2} + 776108 p^{7} T^{3} + p^{14} T^{4}
41D4D_{4} 1+935222T+600319227794T2+935222p7T3+p14T4 1 + 935222 T + 600319227794 T^{2} + 935222 p^{7} T^{3} + p^{14} T^{4}
43D4D_{4} 1+727354T+506716055022T2+727354p7T3+p14T4 1 + 727354 T + 506716055022 T^{2} + 727354 p^{7} T^{3} + p^{14} T^{4}
47D4D_{4} 1+916211T+1020403468094T2+916211p7T3+p14T4 1 + 916211 T + 1020403468094 T^{2} + 916211 p^{7} T^{3} + p^{14} T^{4}
53D4D_{4} 1−29606T−543261749398T2−29606p7T3+p14T4 1 - 29606 T - 543261749398 T^{2} - 29606 p^{7} T^{3} + p^{14} T^{4}
59D4D_{4} 1−1596008T+5092471155638T2−1596008p7T3+p14T4 1 - 1596008 T + 5092471155638 T^{2} - 1596008 p^{7} T^{3} + p^{14} T^{4}
61D4D_{4} 1−109862T+6102866214282T2−109862p7T3+p14T4 1 - 109862 T + 6102866214282 T^{2} - 109862 p^{7} T^{3} + p^{14} T^{4}
67D4D_{4} 1−4832652T+17896221486422T2−4832652p7T3+p14T4 1 - 4832652 T + 17896221486422 T^{2} - 4832652 p^{7} T^{3} + p^{14} T^{4}
71D4D_{4} 1+1051952T+18266757793454T2+1051952p7T3+p14T4 1 + 1051952 T + 18266757793454 T^{2} + 1051952 p^{7} T^{3} + p^{14} T^{4}
73D4D_{4} 1+3819128T+23764134405534T2+3819128p7T3+p14T4 1 + 3819128 T + 23764134405534 T^{2} + 3819128 p^{7} T^{3} + p^{14} T^{4}
79D4D_{4} 1−6266441T+25936515528558T2−6266441p7T3+p14T4 1 - 6266441 T + 25936515528558 T^{2} - 6266441 p^{7} T^{3} + p^{14} T^{4}
83D4D_{4} 1+15296600T+107991709313990T2+15296600p7T3+p14T4 1 + 15296600 T + 107991709313990 T^{2} + 15296600 p^{7} T^{3} + p^{14} T^{4}
89D4D_{4} 1−1772750T+12446633718458T2−1772750p7T3+p14T4 1 - 1772750 T + 12446633718458 T^{2} - 1772750 p^{7} T^{3} + p^{14} T^{4}
97D4D_{4} 1+5483489T+115007232513276T2+5483489p7T3+p14T4 1 + 5483489 T + 115007232513276 T^{2} + 5483489 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

−13.10609758522797859771256821079, −12.42206162447943073109168328261, −11.58563045935552238804058549910, −11.22657382225733352169505231375, −10.29298500099118194441680217492, −10.14592780685252230672728943790, −9.327596019227754802425382397941, −9.024684510819137764085553368750, −8.324708806454048781551817782678, −8.180956523572888834329760375498, −6.98673042660931856733980110776, −6.75315662679899539543134748377, −5.57794565451507156139192684169, −5.43569235955916930903886967872, −3.41538613585248844474751170471, −3.19234029650960505055856048384, −2.08826371578148989409550834858, −1.80203597040632905700581340077, 0, 0, 1.80203597040632905700581340077, 2.08826371578148989409550834858, 3.19234029650960505055856048384, 3.41538613585248844474751170471, 5.43569235955916930903886967872, 5.57794565451507156139192684169, 6.75315662679899539543134748377, 6.98673042660931856733980110776, 8.180956523572888834329760375498, 8.324708806454048781551817782678, 9.024684510819137764085553368750, 9.327596019227754802425382397941, 10.14592780685252230672728943790, 10.29298500099118194441680217492, 11.22657382225733352169505231375, 11.58563045935552238804058549910, 12.42206162447943073109168328261, 13.10609758522797859771256821079

Graph of the ZZ-function along the critical line