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Properties

Label 4-10e2-1.1-c31e2-0-1
Degree 44
Conductor 100100
Sign 11
Analytic cond. 3706.023706.02
Root an. cond. 7.802377.80237
Motivic weight 3131
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  − 6.55e4·2-s − 5.90e6·3-s + 3.22e9·4-s + 6.10e10·5-s + 3.86e11·6-s + 1.80e13·7-s − 1.40e14·8-s − 6.39e14·9-s − 4.00e15·10-s − 1.74e16·11-s − 1.90e16·12-s − 2.54e16·13-s − 1.18e18·14-s − 3.60e17·15-s + 5.76e18·16-s − 1.63e19·17-s + 4.19e19·18-s − 1.07e20·19-s + 1.96e20·20-s − 1.06e20·21-s + 1.14e21·22-s − 5.77e20·23-s + 8.31e20·24-s + 2.79e21·25-s + 1.67e21·26-s + 4.11e21·27-s + 5.80e22·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.237·3-s + 3/2·4-s + 0.894·5-s + 0.335·6-s + 1.43·7-s − 1.41·8-s − 1.03·9-s − 1.26·10-s − 1.25·11-s − 0.356·12-s − 0.138·13-s − 2.02·14-s − 0.212·15-s + 5/4·16-s − 1.38·17-s + 1.46·18-s − 1.63·19-s + 1.34·20-s − 0.340·21-s + 1.77·22-s − 0.451·23-s + 0.335·24-s + 3/5·25-s + 0.195·26-s + 0.268·27-s + 2.15·28-s + ⋯

Functional equation

Λ(s)=(100s/2 ΓC(s)2 L(s)=( Λ(32−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}
Λ(s)=(100s/2 ΓC(s+31/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+31/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 100100    =    22⋅522^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 3706.023706.02
Root analytic conductor: 7.802377.80237
Motivic weight: 3131
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 100, ( :31/2,31/2), 1)(4,\ 100,\ (\ :31/2, 31/2),\ 1)

Particular Values

L(16)L(16) == 00
L(12)L(\frac12) == 00
L(332)L(\frac{33}{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+p15T)2 ( 1 + p^{15} T )^{2}
5C1C_1 (1−p15T)2 ( 1 - p^{15} T )^{2}
good3D4D_{4} 1+656084p2T+308501155394p7T2+656084p33T3+p62T4 1 + 656084 p^{2} T + 308501155394 p^{7} T^{2} + 656084 p^{33} T^{3} + p^{62} T^{4}
7D4D_{4} 1−2573795013316pT+ 1 - 2573795013316 p T + 13 ⁣⋯ ⁣2213\!\cdots\!22p4T2−2573795013316p32T3+p62T4 p^{4} T^{2} - 2573795013316 p^{32} T^{3} + p^{62} T^{4}
11D4D_{4} 1+17416121431336176T+ 1 + 17416121431336176 T + 35 ⁣⋯ ⁣4635\!\cdots\!46p2T2+17416121431336176p31T3+p62T4 p^{2} T^{2} + 17416121431336176 p^{31} T^{3} + p^{62} T^{4}
13D4D_{4} 1+1960754787185492pT+ 1 + 1960754787185492 p T + 23 ⁣⋯ ⁣7423\!\cdots\!74p3T2+1960754787185492p32T3+p62T4 p^{3} T^{2} + 1960754787185492 p^{32} T^{3} + p^{62} T^{4}
17D4D_{4} 1+16396230747963647148T+ 1 + 16396230747963647148 T + 13 ⁣⋯ ⁣2613\!\cdots\!26pT2+16396230747963647148p31T3+p62T4 p T^{2} + 16396230747963647148 p^{31} T^{3} + p^{62} T^{4}
19D4D_{4} 1+5677916680435004360pT+ 1 + 5677916680435004360 p T + 21 ⁣⋯ ⁣5821\!\cdots\!58p2T2+5677916680435004360p32T3+p62T4 p^{2} T^{2} + 5677916680435004360 p^{32} T^{3} + p^{62} T^{4}
23D4D_{4} 1+ 1 + 57 ⁣⋯ ⁣3657\!\cdots\!36T− T - 61 ⁣⋯ ⁣1461\!\cdots\!14pT2+ p T^{2} + 57 ⁣⋯ ⁣3657\!\cdots\!36p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
29D4D_{4} 1− 1 - 67 ⁣⋯ ⁣4067\!\cdots\!40T+ T + 15 ⁣⋯ ⁣0215\!\cdots\!02pT2− p T^{2} - 67 ⁣⋯ ⁣4067\!\cdots\!40p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
31D4D_{4} 1− 1 - 56 ⁣⋯ ⁣6456\!\cdots\!64T+ T + 21 ⁣⋯ ⁣8621\!\cdots\!86T2− T^{2} - 56 ⁣⋯ ⁣6456\!\cdots\!64p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
37D4D_{4} 1− 1 - 28 ⁣⋯ ⁣3228\!\cdots\!32T+ T + 10 ⁣⋯ ⁣8210\!\cdots\!82T2− T^{2} - 28 ⁣⋯ ⁣3228\!\cdots\!32p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
41D4D_{4} 1− 1 - 18 ⁣⋯ ⁣8418\!\cdots\!84T+ T + 14 ⁣⋯ ⁣4614\!\cdots\!46T2− T^{2} - 18 ⁣⋯ ⁣8418\!\cdots\!84p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
43D4D_{4} 1− 1 - 25 ⁣⋯ ⁣8425\!\cdots\!84T+ T + 20 ⁣⋯ ⁣4620\!\cdots\!46pT2− p T^{2} - 25 ⁣⋯ ⁣8425\!\cdots\!84p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
47D4D_{4} 1+ 1 + 20 ⁣⋯ ⁣2820\!\cdots\!28T+ T + 20 ⁣⋯ ⁣0220\!\cdots\!02T2+ T^{2} + 20 ⁣⋯ ⁣2820\!\cdots\!28p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
53D4D_{4} 1− 1 - 93 ⁣⋯ ⁣4493\!\cdots\!44T+ T + 56 ⁣⋯ ⁣7856\!\cdots\!78T2− T^{2} - 93 ⁣⋯ ⁣4493\!\cdots\!44p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
59D4D_{4} 1− 1 - 68 ⁣⋯ ⁣2068\!\cdots\!20pT+ p T + 12 ⁣⋯ ⁣1812\!\cdots\!18T2− T^{2} - 68 ⁣⋯ ⁣2068\!\cdots\!20p32T3+p62T4 p^{32} T^{3} + p^{62} T^{4}
61D4D_{4} 1+ 1 + 52 ⁣⋯ ⁣7652\!\cdots\!76T+ T + 39 ⁣⋯ ⁣6639\!\cdots\!66T2+ T^{2} + 52 ⁣⋯ ⁣7652\!\cdots\!76p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
67D4D_{4} 1− 1 - 35 ⁣⋯ ⁣5235\!\cdots\!52T+ T + 23 ⁣⋯ ⁣4223\!\cdots\!42T2− T^{2} - 35 ⁣⋯ ⁣5235\!\cdots\!52p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
71D4D_{4} 1+ 1 + 15 ⁣⋯ ⁣5615\!\cdots\!56T+ T + 10 ⁣⋯ ⁣2610\!\cdots\!26T2+ T^{2} + 15 ⁣⋯ ⁣5615\!\cdots\!56p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
73D4D_{4} 1+ 1 + 12 ⁣⋯ ⁣3612\!\cdots\!36T+ T + 15 ⁣⋯ ⁣7815\!\cdots\!78T2+ T^{2} + 12 ⁣⋯ ⁣3612\!\cdots\!36p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
79D4D_{4} 1+ 1 + 10 ⁣⋯ ⁣6010\!\cdots\!60T+ T + 11 ⁣⋯ ⁣5811\!\cdots\!58T2+ T^{2} + 10 ⁣⋯ ⁣6010\!\cdots\!60p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
83D4D_{4} 1+ 1 + 16 ⁣⋯ ⁣7616\!\cdots\!76T+ T + 62 ⁣⋯ ⁣7862\!\cdots\!78T2+ T^{2} + 16 ⁣⋯ ⁣7616\!\cdots\!76p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
89D4D_{4} 1− 1 - 59 ⁣⋯ ⁣2059\!\cdots\!20T+ T + 48 ⁣⋯ ⁣7848\!\cdots\!78T2− T^{2} - 59 ⁣⋯ ⁣2059\!\cdots\!20p31T3+p62T4 p^{31} T^{3} + p^{62} T^{4}
97D4D_{4} 1− 1 - 83 ⁣⋯ ⁣7283\!\cdots\!72T+ T + 93 ⁣⋯ ⁣0293\!\cdots\!02T2− T^{2} - 83 ⁣⋯ ⁣7283\!\cdots\!72p31T3+p62T4 p^{31} T^{3} + p^{62} 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

−13.36071533348091776094050780466, −12.77433843909785573955697051680, −11.47537877094027112145931539672, −11.40245634256205334941756820671, −10.45006599346610319078834593893, −10.26924689275884396820989377421, −9.131936585573792538904487484507, −8.481301101531445725476991858886, −8.213945457805781123827835595005, −7.41210979656458481029895846295, −6.25248415101165197798687948959, −6.08891750021286976143166536811, −4.95846014527165799493772555322, −4.49057038408816486893496607961, −2.73285973627266801187767919575, −2.50451788042048994607758862042, −1.81074769645060792390397805515, −1.15784275083442248784408547920, 0, 0, 1.15784275083442248784408547920, 1.81074769645060792390397805515, 2.50451788042048994607758862042, 2.73285973627266801187767919575, 4.49057038408816486893496607961, 4.95846014527165799493772555322, 6.08891750021286976143166536811, 6.25248415101165197798687948959, 7.41210979656458481029895846295, 8.213945457805781123827835595005, 8.481301101531445725476991858886, 9.131936585573792538904487484507, 10.26924689275884396820989377421, 10.45006599346610319078834593893, 11.40245634256205334941756820671, 11.47537877094027112145931539672, 12.77433843909785573955697051680, 13.36071533348091776094050780466

Graph of the ZZ-function along the critical line