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Properties

Label 4-2e2-1.1-c41e2-0-0
Degree 44
Conductor 44
Sign 11
Analytic cond. 453.448453.448
Root an. cond. 4.614574.61457
Motivic weight 4141
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.09e6·2-s + 8.86e9·3-s + 3.29e12·4-s + 9.75e13·5-s + 1.85e16·6-s + 2.17e17·7-s + 4.61e18·8-s + 2.37e19·9-s + 2.04e20·10-s + 1.10e20·11-s + 2.92e22·12-s + 1.73e23·13-s + 4.55e23·14-s + 8.65e23·15-s + 6.04e24·16-s − 1.02e25·17-s + 4.98e25·18-s + 2.06e26·19-s + 3.21e26·20-s + 1.92e27·21-s + 2.31e26·22-s + 1.49e28·23-s + 4.08e28·24-s + 8.05e27·25-s + 3.63e29·26-s + 4.87e28·27-s + 7.16e29·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.46·3-s + 3/2·4-s + 0.457·5-s + 2.07·6-s + 1.02·7-s + 1.41·8-s + 0.652·9-s + 0.647·10-s + 0.0495·11-s + 2.20·12-s + 2.52·13-s + 1.45·14-s + 0.671·15-s + 5/4·16-s − 0.613·17-s + 0.922·18-s + 1.25·19-s + 0.686·20-s + 1.50·21-s + 0.0700·22-s + 1.81·23-s + 2.07·24-s + 0.177·25-s + 3.57·26-s + 0.221·27-s + 1.54·28-s + ⋯

Functional equation

Λ(s)=(4s/2 ΓC(s)2 L(s)=( Λ(42−s)\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}
Λ(s)=(4s/2 ΓC(s+41/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+41/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 44    =    222^{2}
Sign: 11
Analytic conductor: 453.448453.448
Root analytic conductor: 4.614574.61457
Motivic weight: 4141
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 4, ( :41/2,41/2), 1)(4,\ 4,\ (\ :41/2, 41/2),\ 1)

Particular Values

L(21)L(21) ≈\approx 18.1697929218.16979292
L(12)L(\frac12) ≈\approx 18.1697929218.16979292
L(432)L(\frac{43}{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−p20T)2 ( 1 - p^{20} T )^{2}
good3D4D_{4} 1−984816392p2T+927466955744998p10T2−984816392p43T3+p82T4 1 - 984816392 p^{2} T + 927466955744998 p^{10} T^{2} - 984816392 p^{43} T^{3} + p^{82} T^{4}
5D4D_{4} 1−19519836865116pT+ 1 - 19519836865116 p T + 47 ⁣⋯ ⁣4647\!\cdots\!46p5T2−19519836865116p42T3+p82T4 p^{5} T^{2} - 19519836865116 p^{42} T^{3} + p^{82} T^{4}
7D4D_{4} 1−4430751068838544p2T+ 1 - 4430751068838544 p^{2} T + 79 ⁣⋯ ⁣8679\!\cdots\!86p7T2−4430751068838544p43T3+p82T4 p^{7} T^{2} - 4430751068838544 p^{43} T^{3} + p^{82} T^{4}
11D4D_{4} 1−10054043771358142344pT+ 1 - 10054043771358142344 p T + 64 ⁣⋯ ⁣0664\!\cdots\!06p3T2−10054043771358142344p42T3+p82T4 p^{3} T^{2} - 10054043771358142344 p^{42} T^{3} + p^{82} T^{4}
13D4D_{4} 1− 1 - 13 ⁣⋯ ⁣3613\!\cdots\!36pT+ p T + 76 ⁣⋯ ⁣0676\!\cdots\!06p3T2− p^{3} T^{2} - 13 ⁣⋯ ⁣3613\!\cdots\!36p42T3+p82T4 p^{42} T^{3} + p^{82} T^{4}
17D4D_{4} 1+ 1 + 10 ⁣⋯ ⁣0410\!\cdots\!04T+ T + 34 ⁣⋯ ⁣1434\!\cdots\!14pT2+ p T^{2} + 10 ⁣⋯ ⁣0410\!\cdots\!04p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
19D4D_{4} 1− 1 - 10 ⁣⋯ ⁣0010\!\cdots\!00pT+ p T + 65 ⁣⋯ ⁣8265\!\cdots\!82p3T2− p^{3} T^{2} - 10 ⁣⋯ ⁣0010\!\cdots\!00p42T3+p82T4 p^{42} T^{3} + p^{82} T^{4}
23D4D_{4} 1− 1 - 14 ⁣⋯ ⁣6814\!\cdots\!68T+ T + 82 ⁣⋯ ⁣7482\!\cdots\!74pT2− p T^{2} - 14 ⁣⋯ ⁣6814\!\cdots\!68p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
29D4D_{4} 1+ 1 + 47 ⁣⋯ ⁣8047\!\cdots\!80pT+ p T + 47 ⁣⋯ ⁣0247\!\cdots\!02pT2+ p T^{2} + 47 ⁣⋯ ⁣8047\!\cdots\!80p42T3+p82T4 p^{42} T^{3} + p^{82} T^{4}
31D4D_{4} 1+ 1 + 11 ⁣⋯ ⁣3611\!\cdots\!36pT+ p T + 10 ⁣⋯ ⁣6610\!\cdots\!66p2T2+ p^{2} T^{2} + 11 ⁣⋯ ⁣3611\!\cdots\!36p42T3+p82T4 p^{42} T^{3} + p^{82} T^{4}
37D4D_{4} 1− 1 - 32 ⁣⋯ ⁣2832\!\cdots\!28pT+ p T + 31 ⁣⋯ ⁣4231\!\cdots\!42p2T2− p^{2} T^{2} - 32 ⁣⋯ ⁣2832\!\cdots\!28p42T3+p82T4 p^{42} T^{3} + p^{82} T^{4}
41D4D_{4} 1− 1 - 12 ⁣⋯ ⁣6412\!\cdots\!64T+ T + 75 ⁣⋯ ⁣0675\!\cdots\!06T2− T^{2} - 12 ⁣⋯ ⁣6412\!\cdots\!64p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
43D4D_{4} 1+ 1 + 60 ⁣⋯ ⁣5260\!\cdots\!52T+ T + 20 ⁣⋯ ⁣6220\!\cdots\!62T2+ T^{2} + 60 ⁣⋯ ⁣5260\!\cdots\!52p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
47D4D_{4} 1+ 1 + 25 ⁣⋯ ⁣6425\!\cdots\!64T+ T + 84 ⁣⋯ ⁣1884\!\cdots\!18T2+ T^{2} + 25 ⁣⋯ ⁣6425\!\cdots\!64p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
53D4D_{4} 1+ 1 + 30 ⁣⋯ ⁣5230\!\cdots\!52T+ T + 11 ⁣⋯ ⁣8211\!\cdots\!82T2+ T^{2} + 30 ⁣⋯ ⁣5230\!\cdots\!52p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
59D4D_{4} 1+ 1 + 32 ⁣⋯ ⁣4032\!\cdots\!40T+ T + 78 ⁣⋯ ⁣1878\!\cdots\!18T2+ T^{2} + 32 ⁣⋯ ⁣4032\!\cdots\!40p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
61D4D_{4} 1+ 1 + 44 ⁣⋯ ⁣3644\!\cdots\!36T+ T + 17 ⁣⋯ ⁣4617\!\cdots\!46T2+ T^{2} + 44 ⁣⋯ ⁣3644\!\cdots\!36p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
67D4D_{4} 1− 1 - 55 ⁣⋯ ⁣5655\!\cdots\!56T+ T + 15 ⁣⋯ ⁣1815\!\cdots\!18T2− T^{2} - 55 ⁣⋯ ⁣5655\!\cdots\!56p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
71D4D_{4} 1+ 1 + 48 ⁣⋯ ⁣1648\!\cdots\!16T+ T + 10 ⁣⋯ ⁣0610\!\cdots\!06T2+ T^{2} + 48 ⁣⋯ ⁣1648\!\cdots\!16p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
73D4D_{4} 1+ 1 + 48 ⁣⋯ ⁣1248\!\cdots\!12T+ T + 10 ⁣⋯ ⁣8210\!\cdots\!82T2+ T^{2} + 48 ⁣⋯ ⁣1248\!\cdots\!12p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
79D4D_{4} 1+ 1 + 90 ⁣⋯ ⁣8090\!\cdots\!80T+ T + 14 ⁣⋯ ⁣5814\!\cdots\!58T2+ T^{2} + 90 ⁣⋯ ⁣8090\!\cdots\!80p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
83D4D_{4} 1− 1 - 12 ⁣⋯ ⁣6812\!\cdots\!68T+ T + 95 ⁣⋯ ⁣2295\!\cdots\!22T2− T^{2} - 12 ⁣⋯ ⁣6812\!\cdots\!68p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
89D4D_{4} 1− 1 - 26 ⁣⋯ ⁣8026\!\cdots\!80T+ T + 16 ⁣⋯ ⁣7816\!\cdots\!78T2− T^{2} - 26 ⁣⋯ ⁣8026\!\cdots\!80p41T3+p82T4 p^{41} T^{3} + p^{82} T^{4}
97D4D_{4} 1+ 1 + 31 ⁣⋯ ⁣0431\!\cdots\!04T+ T + 59 ⁣⋯ ⁣9859\!\cdots\!98T2+ T^{2} + 31 ⁣⋯ ⁣0431\!\cdots\!04p41T3+p82T4 p^{41} T^{3} + p^{82} 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

−18.78460477151683942421462969779, −17.96160461071481857497799814239, −16.50444697546084319802645030174, −15.57017114748410085567426394887, −14.70036525177306077038356057579, −14.26588627843624105406328000972, −13.28692402703849081191949682565, −13.16628907179816938592848982938, −11.22235630172916294772901008903, −11.12914078299579263768529450063, −9.266134554387374088129282103860, −8.496633578204113835641704222385, −7.59650168024607941287050645931, −6.39130410049944954266522517283, −5.42902154586998137447859334147, −4.46147451943213408737526442809, −3.30484653561010155266951809653, −3.08688946578043229440874758303, −1.68931993327814447396515014980, −1.38155187090491114534548989819, 1.38155187090491114534548989819, 1.68931993327814447396515014980, 3.08688946578043229440874758303, 3.30484653561010155266951809653, 4.46147451943213408737526442809, 5.42902154586998137447859334147, 6.39130410049944954266522517283, 7.59650168024607941287050645931, 8.496633578204113835641704222385, 9.266134554387374088129282103860, 11.12914078299579263768529450063, 11.22235630172916294772901008903, 13.16628907179816938592848982938, 13.28692402703849081191949682565, 14.26588627843624105406328000972, 14.70036525177306077038356057579, 15.57017114748410085567426394887, 16.50444697546084319802645030174, 17.96160461071481857497799814239, 18.78460477151683942421462969779

Graph of the ZZ-function along the critical line