Properties

Label 4-10e2-1.1-c27e2-0-1
Degree 44
Conductor 100100
Sign 11
Analytic cond. 2133.102133.10
Root an. cond. 6.795996.79599
Motivic weight 2727
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  − 1.63e4·2-s − 3.34e6·3-s + 2.01e8·4-s + 2.44e9·5-s + 5.47e10·6-s − 5.87e10·7-s − 2.19e12·8-s − 3.43e12·9-s − 4.00e13·10-s − 1.48e14·11-s − 6.72e14·12-s + 7.00e14·13-s + 9.61e14·14-s − 8.15e15·15-s + 2.25e16·16-s + 3.83e16·17-s + 5.62e16·18-s + 2.31e17·19-s + 4.91e17·20-s + 1.96e17·21-s + 2.43e18·22-s − 1.05e18·23-s + 7.34e18·24-s + 4.47e18·25-s − 1.14e19·26-s + 3.47e19·27-s − 1.18e19·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.20·3-s + 3/2·4-s + 0.894·5-s + 1.71·6-s − 0.229·7-s − 1.41·8-s − 0.450·9-s − 1.26·10-s − 1.29·11-s − 1.81·12-s + 0.641·13-s + 0.323·14-s − 1.08·15-s + 5/4·16-s + 0.939·17-s + 0.636·18-s + 1.26·19-s + 1.34·20-s + 0.277·21-s + 1.83·22-s − 0.435·23-s + 1.71·24-s + 3/5·25-s − 0.906·26-s + 1.64·27-s − 0.343·28-s + ⋯

Functional equation

Λ(s)=(100s/2ΓC(s)2L(s)=(Λ(28s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}
Λ(s)=(100s/2ΓC(s+27/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 100100    =    22522^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 2133.102133.10
Root analytic conductor: 6.795996.79599
Motivic weight: 2727
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 100, ( :27/2,27/2), 1)(4,\ 100,\ (\ :27/2, 27/2),\ 1)

Particular Values

L(14)L(14) == 00
L(12)L(\frac12) == 00
L(292)L(\frac{29}{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)
bad2C1C_1 (1+p13T)2 ( 1 + p^{13} T )^{2}
5C1C_1 (1p13T)2 ( 1 - p^{13} T )^{2}
good3D4D_{4} 1+1113548pT+60050290706p5T2+1113548p28T3+p54T4 1 + 1113548 p T + 60050290706 p^{5} T^{2} + 1113548 p^{28} T^{3} + p^{54} T^{4}
7D4D_{4} 1+8386691756pT+ 1 + 8386691756 p T + 20 ⁣ ⁣9820\!\cdots\!98p2T2+8386691756p28T3+p54T4 p^{2} T^{2} + 8386691756 p^{28} T^{3} + p^{54} T^{4}
11D4D_{4} 1+1227968656176p2T+ 1 + 1227968656176 p^{2} T + 11 ⁣ ⁣6611\!\cdots\!66p3T2+1227968656176p29T3+p54T4 p^{3} T^{2} + 1227968656176 p^{29} T^{3} + p^{54} T^{4}
13D4D_{4} 153853778386892pT+ 1 - 53853778386892 p T + 45 ⁣ ⁣0245\!\cdots\!02p2T253853778386892p28T3+p54T4 p^{2} T^{2} - 53853778386892 p^{28} T^{3} + p^{54} T^{4}
17D4D_{4} 138363966708465748T+ 1 - 38363966708465748 T + 17 ⁣ ⁣6617\!\cdots\!66pT238363966708465748p27T3+p54T4 p T^{2} - 38363966708465748 p^{27} T^{3} + p^{54} T^{4}
19D4D_{4} 112161973113018200pT+ 1 - 12161973113018200 p T + 14 ⁣ ⁣9814\!\cdots\!98p2T212161973113018200p28T3+p54T4 p^{2} T^{2} - 12161973113018200 p^{28} T^{3} + p^{54} T^{4}
23D4D_{4} 1+1051524217202377644T+ 1 + 1051524217202377644 T + 14 ⁣ ⁣8614\!\cdots\!86pT2+1051524217202377644p27T3+p54T4 p T^{2} + 1051524217202377644 p^{27} T^{3} + p^{54} T^{4}
29D4D_{4} 159213295656400885420T+ 1 - 59213295656400885420 T + 63 ⁣ ⁣1863\!\cdots\!18T259213295656400885420p27T3+p54T4 T^{2} - 59213295656400885420 p^{27} T^{3} + p^{54} T^{4}
31D4D_{4} 1 1 - 10 ⁣ ⁣4410\!\cdots\!44T+ T + 27 ⁣ ⁣0627\!\cdots\!06T2 T^{2} - 10 ⁣ ⁣4410\!\cdots\!44p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
37D4D_{4} 1+93534665260879182692T 1 + 93534665260879182692 T - 45 ⁣ ⁣1845\!\cdots\!18T2+93534665260879182692p27T3+p54T4 T^{2} + 93534665260879182692 p^{27} T^{3} + p^{54} T^{4}
41D4D_{4} 1+ 1 + 10 ⁣ ⁣5610\!\cdots\!56T+ T + 95 ⁣ ⁣4695\!\cdots\!46T2+ T^{2} + 10 ⁣ ⁣5610\!\cdots\!56p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
43D4D_{4} 1+ 1 + 47 ⁣ ⁣6447\!\cdots\!64T+ T + 25 ⁣ ⁣3825\!\cdots\!38T2+ T^{2} + 47 ⁣ ⁣6447\!\cdots\!64p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
47D4D_{4} 1+ 1 + 12 ⁣ ⁣7212\!\cdots\!72T+ T + 27 ⁣ ⁣2227\!\cdots\!22T2+ T^{2} + 12 ⁣ ⁣7212\!\cdots\!72p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
53D4D_{4} 1+ 1 + 10 ⁣ ⁣0410\!\cdots\!04T+ T + 45 ⁣ ⁣7845\!\cdots\!78T2+ T^{2} + 10 ⁣ ⁣0410\!\cdots\!04p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
59D4D_{4} 1+ 1 + 18 ⁣ ⁣6018\!\cdots\!60T+ T + 21 ⁣ ⁣3821\!\cdots\!38T2+ T^{2} + 18 ⁣ ⁣6018\!\cdots\!60p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
61D4D_{4} 1+ 1 + 37 ⁣ ⁣1637\!\cdots\!16T+ T + 66 ⁣ ⁣0666\!\cdots\!06T2+ T^{2} + 37 ⁣ ⁣1637\!\cdots\!16p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
67D4D_{4} 1 1 - 37 ⁣ ⁣2837\!\cdots\!28T+ T + 40 ⁣ ⁣4240\!\cdots\!42T2 T^{2} - 37 ⁣ ⁣2837\!\cdots\!28p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
71D4D_{4} 1+ 1 + 48 ⁣ ⁣7648\!\cdots\!76T+ T + 19 ⁣ ⁣2619\!\cdots\!26T2+ T^{2} + 48 ⁣ ⁣7648\!\cdots\!76p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
73D4D_{4} 1 1 - 94 ⁣ ⁣9694\!\cdots\!96T+ T + 32 ⁣ ⁣9832\!\cdots\!98T2 T^{2} - 94 ⁣ ⁣9694\!\cdots\!96p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
79D4D_{4} 1 1 - 91 ⁣ ⁣8091\!\cdots\!80T+ T + 42 ⁣ ⁣4242\!\cdots\!42pT2 p T^{2} - 91 ⁣ ⁣8091\!\cdots\!80p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
83D4D_{4} 1 1 - 14 ⁣ ⁣1614\!\cdots\!16T+ T + 11 ⁣ ⁣1811\!\cdots\!18T2 T^{2} - 14 ⁣ ⁣1614\!\cdots\!16p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
89D4D_{4} 1+ 1 + 16 ⁣ ⁣8016\!\cdots\!80T+ T + 92 ⁣ ⁣5892\!\cdots\!58T2+ T^{2} + 16 ⁣ ⁣8016\!\cdots\!80p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
97D4D_{4} 1+ 1 + 88 ⁣ ⁣3688\!\cdots\!36pT+ p T + 10 ⁣ ⁣4210\!\cdots\!42T2+ T^{2} + 88 ⁣ ⁣3688\!\cdots\!36p28T3+p54T4 p^{28} T^{3} + p^{54} 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

−13.97756821922551499166121253593, −13.49467399399764701293582925178, −12.08816433248692243662481854452, −12.07522271487377906900866341107, −10.82351813602922164737248597228, −10.72478428543871405819678904805, −9.840799220441137020884259230077, −9.349324796154985816624834683360, −8.190493865173437379062652528635, −7.926125608269574303601018424469, −6.61010164811198320189881389167, −6.23138509488249428180098377454, −5.44049308993059938142364929025, −5.03604025707529726661101869382, −3.05300331982851093961720436755, −2.88574255606246008863666505496, −1.56573529971417964589857727623, −1.13126610315316071062535865744, 0, 0, 1.13126610315316071062535865744, 1.56573529971417964589857727623, 2.88574255606246008863666505496, 3.05300331982851093961720436755, 5.03604025707529726661101869382, 5.44049308993059938142364929025, 6.23138509488249428180098377454, 6.61010164811198320189881389167, 7.926125608269574303601018424469, 8.190493865173437379062652528635, 9.349324796154985816624834683360, 9.840799220441137020884259230077, 10.72478428543871405819678904805, 10.82351813602922164737248597228, 12.07522271487377906900866341107, 12.08816433248692243662481854452, 13.49467399399764701293582925178, 13.97756821922551499166121253593

Graph of the ZZ-function along the critical line