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Properties

Label 4-45e2-1.1-c4e2-0-1
Degree 44
Conductor 20252025
Sign 11
Analytic cond. 21.637821.6378
Root an. cond. 2.156762.15676
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  − 10·2-s + 50·4-s + 50·5-s + 80·7-s − 160·8-s − 500·10-s − 200·11-s + 410·13-s − 800·14-s + 444·16-s + 470·17-s + 2.50e3·20-s + 2.00e3·22-s + 680·23-s + 1.87e3·25-s − 4.10e3·26-s + 4.00e3·28-s + 856·31-s − 1.88e3·32-s − 4.70e3·34-s + 4.00e3·35-s − 1.51e3·37-s − 8.00e3·40-s + 1.90e3·41-s − 2.44e3·43-s − 1.00e4·44-s − 6.80e3·46-s + ⋯
L(s)  = 1  − 5/2·2-s + 25/8·4-s + 2·5-s + 1.63·7-s − 5/2·8-s − 5·10-s − 1.65·11-s + 2.42·13-s − 4.08·14-s + 1.73·16-s + 1.62·17-s + 25/4·20-s + 4.13·22-s + 1.28·23-s + 3·25-s − 6.06·26-s + 5.10·28-s + 0.890·31-s − 1.83·32-s − 4.06·34-s + 3.26·35-s − 1.10·37-s − 5·40-s + 1.13·41-s − 1.31·43-s − 5.16·44-s − 3.21·46-s + ⋯

Functional equation

Λ(s)=(2025s/2 ΓC(s)2 L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
Λ(s)=(2025s/2 ΓC(s+2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 20252025    =    34⋅523^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 21.637821.6378
Root analytic conductor: 2.156762.15676
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2025, ( :2,2), 1)(4,\ 2025,\ (\ :2, 2),\ 1)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.0258639561.025863956
L(12)L(\frac12) ≈\approx 1.0258639561.025863956
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)
bad3 1 1
5C1C_1 (1−p2T)2 ( 1 - p^{2} T )^{2}
good2C22C_2^2 1+5pT+25pT2+5p5T3+p8T4 1 + 5 p T + 25 p T^{2} + 5 p^{5} T^{3} + p^{8} T^{4}
7C22C_2^2 1−80T+3200T2−80p4T3+p8T4 1 - 80 T + 3200 T^{2} - 80 p^{4} T^{3} + p^{8} T^{4}
11C2C_2 (1+100T+p4T2)2 ( 1 + 100 T + p^{4} T^{2} )^{2}
13C22C_2^2 1−410T+84050T2−410p4T3+p8T4 1 - 410 T + 84050 T^{2} - 410 p^{4} T^{3} + p^{8} T^{4}
17C22C_2^2 1−470T+110450T2−470p4T3+p8T4 1 - 470 T + 110450 T^{2} - 470 p^{4} T^{3} + p^{8} T^{4}
19C22C_2^2 1−255458T2+p8T4 1 - 255458 T^{2} + p^{8} T^{4}
23C22C_2^2 1−680T+231200T2−680p4T3+p8T4 1 - 680 T + 231200 T^{2} - 680 p^{4} T^{3} + p^{8} T^{4}
29C22C_2^2 1−1212062T2+p8T4 1 - 1212062 T^{2} + p^{8} T^{4}
31C2C_2 (1−428T+p4T2)2 ( 1 - 428 T + p^{4} T^{2} )^{2}
37C22C_2^2 1+1510T+1140050T2+1510p4T3+p8T4 1 + 1510 T + 1140050 T^{2} + 1510 p^{4} T^{3} + p^{8} T^{4}
41C2C_2 (1−950T+p4T2)2 ( 1 - 950 T + p^{4} T^{2} )^{2}
43C22C_2^2 1+2440T+2976800T2+2440p4T3+p8T4 1 + 2440 T + 2976800 T^{2} + 2440 p^{4} T^{3} + p^{8} T^{4}
47C22C_2^2 1+640T+204800T2+640p4T3+p8T4 1 + 640 T + 204800 T^{2} + 640 p^{4} T^{3} + p^{8} T^{4}
53C22C_2^2 1−1010T+510050T2−1010p4T3+p8T4 1 - 1010 T + 510050 T^{2} - 1010 p^{4} T^{3} + p^{8} T^{4}
59C22C_2^2 1+15455278T2+p8T4 1 + 15455278 T^{2} + p^{8} T^{4}
61C2C_2 (1+3808T+p4T2)2 ( 1 + 3808 T + p^{4} T^{2} )^{2}
67C22C_2^2 1−680T+231200T2−680p4T3+p8T4 1 - 680 T + 231200 T^{2} - 680 p^{4} T^{3} + p^{8} T^{4}
71C2C_2 (1+3400T+p4T2)2 ( 1 + 3400 T + p^{4} T^{2} )^{2}
73C22C_2^2 1−830T+344450T2−830p4T3+p8T4 1 - 830 T + 344450 T^{2} - 830 p^{4} T^{3} + p^{8} T^{4}
79C22C_2^2 1−32580338T2+p8T4 1 - 32580338 T^{2} + p^{8} T^{4}
83C22C_2^2 1+1360T+924800T2+1360p4T3+p8T4 1 + 1360 T + 924800 T^{2} + 1360 p^{4} T^{3} + p^{8} T^{4}
89C22C_2^2 1−120421982T2+p8T4 1 - 120421982 T^{2} + p^{8} T^{4}
97C22C_2^2 1−3230T+5216450T2−3230p4T3+p8T4 1 - 3230 T + 5216450 T^{2} - 3230 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

−15.60902321995802157718247358034, −14.93335667345749484841391268839, −14.22887761342581863350793123036, −13.60469477727313725774403377547, −13.23243016215544772796885912275, −12.32798336888523913000149918694, −11.13863414495670208643152394214, −10.87071480473455952488790464967, −10.27816425535131573361315562552, −10.05764281745735347802030116292, −8.978313391807621173776114886713, −8.842516389332419937877742541108, −8.063258574828759974350640315673, −7.74570646486791488147425204999, −6.56798480812039893955511375980, −5.61584908961168104185922907072, −5.18008670430117112402853184422, −2.88303186380730606771898343364, −1.39363567087694261239185012619, −1.29531835230208170270457325946, 1.29531835230208170270457325946, 1.39363567087694261239185012619, 2.88303186380730606771898343364, 5.18008670430117112402853184422, 5.61584908961168104185922907072, 6.56798480812039893955511375980, 7.74570646486791488147425204999, 8.063258574828759974350640315673, 8.842516389332419937877742541108, 8.978313391807621173776114886713, 10.05764281745735347802030116292, 10.27816425535131573361315562552, 10.87071480473455952488790464967, 11.13863414495670208643152394214, 12.32798336888523913000149918694, 13.23243016215544772796885912275, 13.60469477727313725774403377547, 14.22887761342581863350793123036, 14.93335667345749484841391268839, 15.60902321995802157718247358034

Graph of the ZZ-function along the critical line