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Properties

Label 4-10e2-1.1-c4e2-0-1
Degree 44
Conductor 100100
Sign 11
Analytic cond. 1.068531.06853
Root an. cond. 1.016711.01671
Motivic weight 44
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  + 4·2-s + 2·3-s + 8·4-s − 30·5-s + 8·6-s − 38·7-s + 2·9-s − 120·10-s + 404·11-s + 16·12-s − 198·13-s − 152·14-s − 60·15-s − 64·16-s − 478·17-s + 8·18-s − 240·20-s − 76·21-s + 1.61e3·22-s + 1.08e3·23-s + 275·25-s − 792·26-s + 162·27-s − 304·28-s − 240·30-s − 1.51e3·31-s − 256·32-s + ⋯
L(s)  = 1  + 2-s + 2/9·3-s + 1/2·4-s − 6/5·5-s + 2/9·6-s − 0.775·7-s + 2/81·9-s − 6/5·10-s + 3.33·11-s + 1/9·12-s − 1.17·13-s − 0.775·14-s − 0.266·15-s − 1/4·16-s − 1.65·17-s + 2/81·18-s − 3/5·20-s − 0.172·21-s + 3.33·22-s + 2.04·23-s + 0.439·25-s − 1.17·26-s + 2/9·27-s − 0.387·28-s − 0.266·30-s − 1.57·31-s − 1/4·32-s + ⋯

Functional equation

Λ(s)=(100s/2 ΓC(s)2 L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
Λ(s)=(100s/2 ΓC(s+2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+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: 1.068531.06853
Root analytic conductor: 1.016711.01671
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 100, ( :2,2), 1)(4,\ 100,\ (\ :2, 2),\ 1)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.4874057461.487405746
L(12)L(\frac12) ≈\approx 1.4874057461.487405746
L(3)L(3) 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)
bad2C2C_2 1−p2T+p3T2 1 - p^{2} T + p^{3} T^{2}
5C2C_2 1+6pT+p4T2 1 + 6 p T + p^{4} T^{2}
good3C22C_2^2 1−2T+2T2−2p4T3+p8T4 1 - 2 T + 2 T^{2} - 2 p^{4} T^{3} + p^{8} T^{4}
7C22C_2^2 1+38T+722T2+38p4T3+p8T4 1 + 38 T + 722 T^{2} + 38 p^{4} T^{3} + p^{8} T^{4}
11C2C_2 (1−202T+p4T2)2 ( 1 - 202 T + p^{4} T^{2} )^{2}
13C22C_2^2 1+198T+19602T2+198p4T3+p8T4 1 + 198 T + 19602 T^{2} + 198 p^{4} T^{3} + p^{8} T^{4}
17C22C_2^2 1+478T+114242T2+478p4T3+p8T4 1 + 478 T + 114242 T^{2} + 478 p^{4} T^{3} + p^{8} T^{4}
19C22C_2^2 1−259042T2+p8T4 1 - 259042 T^{2} + p^{8} T^{4}
23C22C_2^2 1−1082T+585362T2−1082p4T3+p8T4 1 - 1082 T + 585362 T^{2} - 1082 p^{4} T^{3} + p^{8} T^{4}
29C22C_2^2 1−1374562T2+p8T4 1 - 1374562 T^{2} + p^{8} T^{4}
31C2C_2 (1+758T+p4T2)2 ( 1 + 758 T + p^{4} T^{2} )^{2}
37C22C_2^2 1−282T+39762T2−282p4T3+p8T4 1 - 282 T + 39762 T^{2} - 282 p^{4} T^{3} + p^{8} T^{4}
41C2C_2 (1−1042T+p4T2)2 ( 1 - 1042 T + p^{4} T^{2} )^{2}
43C22C_2^2 1+1518T+1152162T2+1518p4T3+p8T4 1 + 1518 T + 1152162 T^{2} + 1518 p^{4} T^{3} + p^{8} T^{4}
47C22C_2^2 1+918T+421362T2+918p4T3+p8T4 1 + 918 T + 421362 T^{2} + 918 p^{4} T^{3} + p^{8} T^{4}
53C22C_2^2 1+3638T+6617522T2+3638p4T3+p8T4 1 + 3638 T + 6617522 T^{2} + 3638 p^{4} T^{3} + p^{8} T^{4}
59C22C_2^2 1−3074722T2+p8T4 1 - 3074722 T^{2} + p^{8} T^{4}
61C2C_2 (1−2082T+p4T2)2 ( 1 - 2082 T + p^{4} T^{2} )^{2}
67C22C_2^2 1−10162T+51633122T2−10162p4T3+p8T4 1 - 10162 T + 51633122 T^{2} - 10162 p^{4} T^{3} + p^{8} T^{4}
71C2C_2 (1+3478T+p4T2)2 ( 1 + 3478 T + p^{4} T^{2} )^{2}
73C22C_2^2 1+6958T+24206882T2+6958p4T3+p8T4 1 + 6958 T + 24206882 T^{2} + 6958 p^{4} T^{3} + p^{8} T^{4}
79C22C_2^2 1−18917762T2+p8T4 1 - 18917762 T^{2} + p^{8} T^{4}
83C22C_2^2 1−12162T+73957122T2−12162p4T3+p8T4 1 - 12162 T + 73957122 T^{2} - 12162 p^{4} T^{3} + p^{8} T^{4}
89C22C_2^2 1−93222082T2+p8T4 1 - 93222082 T^{2} + p^{8} T^{4}
97C22C_2^2 1−1122T+629442T2−1122p4T3+p8T4 1 - 1122 T + 629442 T^{2} - 1122 p^{4} T^{3} + p^{8} 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

−20.23607954311365487738523774895, −20.01403317711729005848201304095, −19.27162279492311257663844232801, −19.20368663275779165457972978862, −17.55502990608387111124039039755, −16.98567584052670781622901093783, −16.26268387309593465847672038068, −15.35825312872810959797410897070, −14.54725242579615035720868542985, −14.50150656506435262624987352586, −13.10819895540757872456905795597, −12.51656295742487448191409429372, −11.61489344213984914249970358162, −11.25713092051544583063154212326, −9.349130222125409586860684392726, −8.963181990116567087645578232943, −7.07316282889632245175698041630, −6.58661284276896890433337774218, −4.48822831160574025841244091289, −3.59263323010505890195366133299, 3.59263323010505890195366133299, 4.48822831160574025841244091289, 6.58661284276896890433337774218, 7.07316282889632245175698041630, 8.963181990116567087645578232943, 9.349130222125409586860684392726, 11.25713092051544583063154212326, 11.61489344213984914249970358162, 12.51656295742487448191409429372, 13.10819895540757872456905795597, 14.50150656506435262624987352586, 14.54725242579615035720868542985, 15.35825312872810959797410897070, 16.26268387309593465847672038068, 16.98567584052670781622901093783, 17.55502990608387111124039039755, 19.20368663275779165457972978862, 19.27162279492311257663844232801, 20.01403317711729005848201304095, 20.23607954311365487738523774895

Graph of the ZZ-function along the critical line