Properties

Label 4-10e2-1.1-c23e2-0-2
Degree 44
Conductor 100100
Sign 11
Analytic cond. 1123.611123.61
Root an. cond. 5.789685.78968
Motivic weight 2323
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  + 4.09e3·2-s − 9.18e4·3-s + 1.25e7·4-s − 9.76e7·5-s − 3.76e8·6-s + 2.14e9·7-s + 3.43e10·8-s − 9.60e10·9-s − 4.00e11·10-s − 6.92e11·11-s − 1.15e12·12-s − 1.70e12·13-s + 8.79e12·14-s + 8.97e12·15-s + 8.79e13·16-s − 9.31e13·17-s − 3.93e14·18-s − 9.70e14·19-s − 1.22e15·20-s − 1.97e14·21-s − 2.83e15·22-s − 9.62e14·23-s − 3.15e15·24-s + 7.15e15·25-s − 6.99e15·26-s + 9.76e15·27-s + 2.70e16·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.299·3-s + 3/2·4-s − 0.894·5-s − 0.423·6-s + 0.410·7-s + 1.41·8-s − 1.01·9-s − 1.26·10-s − 0.731·11-s − 0.449·12-s − 0.264·13-s + 0.580·14-s + 0.267·15-s + 5/4·16-s − 0.659·17-s − 1.44·18-s − 1.91·19-s − 1.34·20-s − 0.122·21-s − 1.03·22-s − 0.210·23-s − 0.423·24-s + 3/5·25-s − 0.373·26-s + 0.338·27-s + 0.615·28-s + ⋯

Functional equation

Λ(s)=(100s/2ΓC(s)2L(s)=(Λ(24s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}
Λ(s)=(100s/2ΓC(s+23/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 100100    =    22522^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 1123.611123.61
Root analytic conductor: 5.789685.78968
Motivic weight: 2323
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 100, ( :23/2,23/2), 1)(4,\ 100,\ (\ :23/2, 23/2),\ 1)

Particular Values

L(12)L(12) == 00
L(12)L(\frac12) == 00
L(252)L(\frac{25}{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 (1p11T)2 ( 1 - p^{11} T )^{2}
5C1C_1 (1+p11T)2 ( 1 + p^{11} T )^{2}
good3D4D_{4} 1+30628pT+429805726p5T2+30628p24T3+p46T4 1 + 30628 p T + 429805726 p^{5} T^{2} + 30628 p^{24} T^{3} + p^{46} T^{4}
7D4D_{4} 1306579844pT+101274343560173214p3T2306579844p24T3+p46T4 1 - 306579844 p T + 101274343560173214 p^{3} T^{2} - 306579844 p^{24} T^{3} + p^{46} T^{4}
11D4D_{4} 1+62935415616pT+ 1 + 62935415616 p T + 15 ⁣ ⁣8615\!\cdots\!86p2T2+62935415616p24T3+p46T4 p^{2} T^{2} + 62935415616 p^{24} T^{3} + p^{46} T^{4}
13D4D_{4} 1+1708452301484T+ 1 + 1708452301484 T + 10 ⁣ ⁣6610\!\cdots\!66pT2+1708452301484p23T3+p46T4 p T^{2} + 1708452301484 p^{23} T^{3} + p^{46} T^{4}
17D4D_{4} 1+93149332240332T+ 1 + 93149332240332 T + 21 ⁣ ⁣4621\!\cdots\!46pT2+93149332240332p23T3+p46T4 p T^{2} + 93149332240332 p^{23} T^{3} + p^{46} T^{4}
19D4D_{4} 1+970438534616600T+ 1 + 970438534616600 T + 38 ⁣ ⁣2238\!\cdots\!22pT2+970438534616600p23T3+p46T4 p T^{2} + 970438534616600 p^{23} T^{3} + p^{46} T^{4}
23D4D_{4} 1+962002498409964T+ 1 + 962002498409964 T + 31 ⁣ ⁣5831\!\cdots\!58T2+962002498409964p23T3+p46T4 T^{2} + 962002498409964 p^{23} T^{3} + p^{46} T^{4}
29D4D_{4} 1+40394756391690180T+ 1 + 40394756391690180 T + 26 ⁣ ⁣7826\!\cdots\!78T2+40394756391690180p23T3+p46T4 T^{2} + 40394756391690180 p^{23} T^{3} + p^{46} T^{4}
31D4D_{4} 1+54819002015519096T+ 1 + 54819002015519096 T + 33 ⁣ ⁣8633\!\cdots\!86T2+54819002015519096p23T3+p46T4 T^{2} + 54819002015519096 p^{23} T^{3} + p^{46} T^{4}
37D4D_{4} 1+2098544042837784332T+ 1 + 2098544042837784332 T + 31 ⁣ ⁣6231\!\cdots\!62T2+2098544042837784332p23T3+p46T4 T^{2} + 2098544042837784332 p^{23} T^{3} + p^{46} T^{4}
41D4D_{4} 1+4273739814696341076T+ 1 + 4273739814696341076 T + 22 ⁣ ⁣8622\!\cdots\!86T2+4273739814696341076p23T3+p46T4 T^{2} + 4273739814696341076 p^{23} T^{3} + p^{46} T^{4}
43D4D_{4} 1+10838321168303261564T+ 1 + 10838321168303261564 T + 99 ⁣ ⁣3899\!\cdots\!38T2+10838321168303261564p23T3+p46T4 T^{2} + 10838321168303261564 p^{23} T^{3} + p^{46} T^{4}
47D4D_{4} 1+26680537462560731892T+ 1 + 26680537462560731892 T + 52 ⁣ ⁣6252\!\cdots\!62T2+26680537462560731892p23T3+p46T4 T^{2} + 26680537462560731892 p^{23} T^{3} + p^{46} T^{4}
53D4D_{4} 133753250226251705956T+ 1 - 33753250226251705956 T + 57 ⁣ ⁣3857\!\cdots\!38T233753250226251705956p23T3+p46T4 T^{2} - 33753250226251705956 p^{23} T^{3} + p^{46} T^{4}
59D4D_{4} 140199521927491576840T+ 1 - 40199521927491576840 T + 66 ⁣ ⁣5866\!\cdots\!58T240199521927491576840p23T3+p46T4 T^{2} - 40199521927491576840 p^{23} T^{3} + p^{46} T^{4}
61D4D_{4} 1 1 - 80 ⁣ ⁣0480\!\cdots\!04T+ T + 37 ⁣ ⁣6637\!\cdots\!66T2 T^{2} - 80 ⁣ ⁣0480\!\cdots\!04p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
67D4D_{4} 1 1 - 21 ⁣ ⁣4821\!\cdots\!48T+ T + 29 ⁣ ⁣0229\!\cdots\!02T2 T^{2} - 21 ⁣ ⁣4821\!\cdots\!48p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
71D4D_{4} 1 1 - 71 ⁣ ⁣6471\!\cdots\!64T T - 15 ⁣ ⁣5415\!\cdots\!54T2 T^{2} - 71 ⁣ ⁣6471\!\cdots\!64p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
73D4D_{4} 1 1 - 31 ⁣ ⁣7631\!\cdots\!76T+ T + 90 ⁣ ⁣7890\!\cdots\!78T2 T^{2} - 31 ⁣ ⁣7631\!\cdots\!76p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
79D4D_{4} 1+ 1 + 70 ⁣ ⁣2070\!\cdots\!20T+ T + 10 ⁣ ⁣7810\!\cdots\!78T2+ T^{2} + 70 ⁣ ⁣2070\!\cdots\!20p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
83D4D_{4} 1+ 1 + 21 ⁣ ⁣4421\!\cdots\!44T+ T + 32 ⁣ ⁣5832\!\cdots\!58T2+ T^{2} + 21 ⁣ ⁣4421\!\cdots\!44p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
89D4D_{4} 1+ 1 + 19 ⁣ ⁣8019\!\cdots\!80T+ T + 11 ⁣ ⁣3811\!\cdots\!38T2+ T^{2} + 19 ⁣ ⁣8019\!\cdots\!80p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
97D4D_{4} 1 1 - 11 ⁣ ⁣8811\!\cdots\!88T+ T + 10 ⁣ ⁣8210\!\cdots\!82T2 T^{2} - 11 ⁣ ⁣8811\!\cdots\!88p23T3+p46T4 p^{23} T^{3} + p^{46} 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

−14.55430448726141694530821031557, −14.52714145053782251705922784833, −13.20883757430877632738312681175, −12.92354923669322687883018516492, −11.95917950019260127343796694263, −11.43494994566668803452833107465, −10.94698648401794219119414465327, −10.13369762318517327652199063516, −8.392302418733245324373918091481, −8.288694230353171644476465211036, −6.94103249727586938170408291690, −6.49645296786283932791073601534, −5.23431630705502910321398833247, −5.06211035121655743552037750147, −3.98167045682760011718907147567, −3.39131263719406303899277868045, −2.39426470800482057139001456287, −1.75844881679893128551930352218, 0, 0, 1.75844881679893128551930352218, 2.39426470800482057139001456287, 3.39131263719406303899277868045, 3.98167045682760011718907147567, 5.06211035121655743552037750147, 5.23431630705502910321398833247, 6.49645296786283932791073601534, 6.94103249727586938170408291690, 8.288694230353171644476465211036, 8.392302418733245324373918091481, 10.13369762318517327652199063516, 10.94698648401794219119414465327, 11.43494994566668803452833107465, 11.95917950019260127343796694263, 12.92354923669322687883018516492, 13.20883757430877632738312681175, 14.52714145053782251705922784833, 14.55430448726141694530821031557

Graph of the ZZ-function along the critical line