Properties

Label 4-10e2-1.1-c23e2-0-1
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 − 1.85e4·3-s + 1.25e7·4-s + 9.76e7·5-s + 7.58e7·6-s − 4.41e9·7-s − 3.43e10·8-s − 1.31e11·9-s − 4.00e11·10-s + 1.95e11·11-s − 2.32e11·12-s + 2.58e12·13-s + 1.80e13·14-s − 1.80e12·15-s + 8.79e13·16-s + 1.31e14·17-s + 5.36e14·18-s − 1.76e12·19-s + 1.22e15·20-s + 8.17e13·21-s − 7.98e14·22-s + 4.01e15·23-s + 6.36e14·24-s + 7.15e15·25-s − 1.05e16·26-s + 3.11e15·27-s − 5.55e16·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.0603·3-s + 3/2·4-s + 0.894·5-s + 0.0853·6-s − 0.844·7-s − 1.41·8-s − 1.39·9-s − 1.26·10-s + 0.206·11-s − 0.0905·12-s + 0.399·13-s + 1.19·14-s − 0.0539·15-s + 5/4·16-s + 0.931·17-s + 1.96·18-s − 0.00346·19-s + 1.34·20-s + 0.0509·21-s − 0.291·22-s + 0.878·23-s + 0.0853·24-s + 3/5·25-s − 0.565·26-s + 0.107·27-s − 1.26·28-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 (1+p11T)2 ( 1 + p^{11} T )^{2}
5C1C_1 (1p11T)2 ( 1 - p^{11} T )^{2}
good3D4D_{4} 1+6172pT+540524626p5T2+6172p24T3+p46T4 1 + 6172 p T + 540524626 p^{5} T^{2} + 6172 p^{24} T^{3} + p^{46} T^{4}
7D4D_{4} 1+4415735108T+990635565626217198p2T2+4415735108p23T3+p46T4 1 + 4415735108 T + 990635565626217198 p^{2} T^{2} + 4415735108 p^{23} T^{3} + p^{46} T^{4}
11D4D_{4} 117728839984pT+ 1 - 17728839984 p T + 57 ⁣ ⁣8657\!\cdots\!86p2T217728839984p24T3+p46T4 p^{2} T^{2} - 17728839984 p^{24} T^{3} + p^{46} T^{4}
13D4D_{4} 12583183058684T+ 1 - 2583183058684 T + 54 ⁣ ⁣6654\!\cdots\!66pT22583183058684p23T3+p46T4 p T^{2} - 2583183058684 p^{23} T^{3} + p^{46} T^{4}
17D4D_{4} 17745227211796pT+ 1 - 7745227211796 p T + 13 ⁣ ⁣3813\!\cdots\!38p2T27745227211796p24T3+p46T4 p^{2} T^{2} - 7745227211796 p^{24} T^{3} + p^{46} T^{4}
19D4D_{4} 1+92678337800pT 1 + 92678337800 p T - 47 ⁣ ⁣6247\!\cdots\!62p2T2+92678337800p24T3+p46T4 p^{2} T^{2} + 92678337800 p^{24} T^{3} + p^{46} T^{4}
23D4D_{4} 14012836105342564T+ 1 - 4012836105342564 T + 41 ⁣ ⁣5841\!\cdots\!58T24012836105342564p23T3+p46T4 T^{2} - 4012836105342564 p^{23} T^{3} + p^{46} T^{4}
29D4D_{4} 1+108385465473787380T+ 1 + 108385465473787380 T + 11 ⁣ ⁣7811\!\cdots\!78T2+108385465473787380p23T3+p46T4 T^{2} + 108385465473787380 p^{23} T^{3} + p^{46} T^{4}
31D4D_{4} 1+282534241930647896T+ 1 + 282534241930647896 T + 58 ⁣ ⁣8658\!\cdots\!86T2+282534241930647896p23T3+p46T4 T^{2} + 282534241930647896 p^{23} T^{3} + p^{46} T^{4}
37D4D_{4} 1+326855967548366468T+ 1 + 326855967548366468 T + 23 ⁣ ⁣6223\!\cdots\!62T2+326855967548366468p23T3+p46T4 T^{2} + 326855967548366468 p^{23} T^{3} + p^{46} T^{4}
41D4D_{4} 1+5378411403014673276T+ 1 + 5378411403014673276 T + 31 ⁣ ⁣8631\!\cdots\!86T2+5378411403014673276p23T3+p46T4 T^{2} + 5378411403014673276 p^{23} T^{3} + p^{46} T^{4}
43D4D_{4} 1+9824206470496391636T+ 1 + 9824206470496391636 T + 93 ⁣ ⁣3893\!\cdots\!38T2+9824206470496391636p23T3+p46T4 T^{2} + 9824206470496391636 p^{23} T^{3} + p^{46} T^{4}
47D4D_{4} 1+18459008132084260308T+ 1 + 18459008132084260308 T + 56 ⁣ ⁣6256\!\cdots\!62T2+18459008132084260308p23T3+p46T4 T^{2} + 18459008132084260308 p^{23} T^{3} + p^{46} T^{4}
53D4D_{4} 1+6735586459267392756T+ 1 + 6735586459267392756 T + 74 ⁣ ⁣3874\!\cdots\!38T2+6735586459267392756p23T3+p46T4 T^{2} + 6735586459267392756 p^{23} T^{3} + p^{46} T^{4}
59D4D_{4} 1 1 - 40 ⁣ ⁣4040\!\cdots\!40T+ T + 14 ⁣ ⁣5814\!\cdots\!58T2 T^{2} - 40 ⁣ ⁣4040\!\cdots\!40p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
61D4D_{4} 1 1 - 26 ⁣ ⁣0426\!\cdots\!04T+ T + 10 ⁣ ⁣6610\!\cdots\!66T2 T^{2} - 26 ⁣ ⁣0426\!\cdots\!04p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
67D4D_{4} 1+ 1 + 87 ⁣ ⁣4887\!\cdots\!48T+ T + 20 ⁣ ⁣0220\!\cdots\!02T2+ T^{2} + 87 ⁣ ⁣4887\!\cdots\!48p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
71D4D_{4} 1+ 1 + 14 ⁣ ⁣3614\!\cdots\!36T+ T + 80 ⁣ ⁣4680\!\cdots\!46T2+ T^{2} + 14 ⁣ ⁣3614\!\cdots\!36p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
73D4D_{4} 1+ 1 + 27 ⁣ ⁣7627\!\cdots\!76T+ T + 64 ⁣ ⁣7864\!\cdots\!78T2+ T^{2} + 27 ⁣ ⁣7627\!\cdots\!76p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
79D4D_{4} 1+ 1 + 16 ⁣ ⁣2016\!\cdots\!20T+ T + 19 ⁣ ⁣8219\!\cdots\!82pT2+ p T^{2} + 16 ⁣ ⁣2016\!\cdots\!20p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
83D4D_{4} 1+ 1 + 24 ⁣ ⁣5624\!\cdots\!56T+ T + 38 ⁣ ⁣5838\!\cdots\!58T2+ T^{2} + 24 ⁣ ⁣5624\!\cdots\!56p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
89D4D_{4} 1+ 1 + 69 ⁣ ⁣8069\!\cdots\!80T+ T + 24 ⁣ ⁣3824\!\cdots\!38T2+ T^{2} + 69 ⁣ ⁣8069\!\cdots\!80p23T3+p46T4 p^{23} T^{3} + p^{46} T^{4}
97D4D_{4} 1 1 - 14 ⁣ ⁣1214\!\cdots\!12T+ T + 12 ⁣ ⁣8212\!\cdots\!82T2 T^{2} - 14 ⁣ ⁣1214\!\cdots\!12p23T3+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

−14.69390718322505392582060265338, −14.58169983205091940843880867931, −13.25939727512564272838476998147, −12.79049491196663800966999533364, −11.42728542323827566773725694286, −11.35150879129843082742683915823, −10.08346713543973632897355434531, −9.828036761207862126136814570754, −8.795024627342551685736053765080, −8.624538790419002043482374226643, −7.35627020794432526751101012849, −6.69201322082391387620331792881, −5.77645227090916990841145681726, −5.40606186739822459580152865245, −3.38027992499909036996118919657, −3.06197370869439677911210663828, −1.87873178612466061232391881201, −1.36549710130965868341011250098, 0, 0, 1.36549710130965868341011250098, 1.87873178612466061232391881201, 3.06197370869439677911210663828, 3.38027992499909036996118919657, 5.40606186739822459580152865245, 5.77645227090916990841145681726, 6.69201322082391387620331792881, 7.35627020794432526751101012849, 8.624538790419002043482374226643, 8.795024627342551685736053765080, 9.828036761207862126136814570754, 10.08346713543973632897355434531, 11.35150879129843082742683915823, 11.42728542323827566773725694286, 12.79049491196663800966999533364, 13.25939727512564272838476998147, 14.58169983205091940843880867931, 14.69390718322505392582060265338

Graph of the ZZ-function along the critical line