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Properties

Label 4-45e2-1.1-c5e2-0-2
Degree 44
Conductor 20252025
Sign 11
Analytic cond. 52.089052.0890
Root an. cond. 2.686492.68649
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 20·4-s + 90·5-s − 504·11-s − 624·16-s − 440·19-s + 1.80e3·20-s + 4.97e3·25-s + 1.38e4·29-s + 1.35e4·31-s + 396·41-s − 1.00e4·44-s + 3.00e4·49-s − 4.53e4·55-s + 4.93e4·59-s − 1.13e4·61-s − 3.29e4·64-s − 1.06e5·71-s − 8.80e3·76-s + 1.03e5·79-s − 5.61e4·80-s + 1.99e4·89-s − 3.96e4·95-s + 9.95e4·100-s + 2.18e5·101-s − 4.20e4·109-s + 2.77e5·116-s − 1.31e5·121-s + ⋯
L(s)  = 1  + 5/8·4-s + 1.60·5-s − 1.25·11-s − 0.609·16-s − 0.279·19-s + 1.00·20-s + 1.59·25-s + 3.06·29-s + 2.52·31-s + 0.0367·41-s − 0.784·44-s + 1.78·49-s − 2.02·55-s + 1.84·59-s − 0.392·61-s − 1.00·64-s − 2.51·71-s − 0.174·76-s + 1.87·79-s − 0.981·80-s + 0.267·89-s − 0.450·95-s + 0.994·100-s + 2.12·101-s − 0.338·109-s + 1.91·116-s − 0.817·121-s + ⋯

Functional equation

Λ(s)=(2025s/2 ΓC(s)2 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(2025s/2 ΓC(s+5/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+5/2)^{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: 52.089052.0890
Root analytic conductor: 2.686492.68649
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2025, ( :5/2,5/2), 1)(4,\ 2025,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.2637282063.263728206
L(12)L(\frac12) ≈\approx 3.2637282063.263728206
L(72)L(\frac{7}{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)
bad3 1 1
5C2C_2 1−18pT+p5T2 1 - 18 p T + p^{5} T^{2}
good2C22C_2^2 1−5p2T2+p10T4 1 - 5 p^{2} T^{2} + p^{10} T^{4}
7C22C_2^2 1−30050T2+p10T4 1 - 30050 T^{2} + p^{10} T^{4}
11C2C_2 (1+252T+p5T2)2 ( 1 + 252 T + p^{5} T^{2} )^{2}
13C22C_2^2 1−728330T2+p10T4 1 - 728330 T^{2} + p^{10} T^{4}
17C22C_2^2 1−2363810T2+p10T4 1 - 2363810 T^{2} + p^{10} T^{4}
19C2C_2 (1+220T+p5T2)2 ( 1 + 220 T + p^{5} T^{2} )^{2}
23C22C_2^2 1−6946370T2+p10T4 1 - 6946370 T^{2} + p^{10} T^{4}
29C2C_2 (1−6930T+p5T2)2 ( 1 - 6930 T + p^{5} T^{2} )^{2}
31C2C_2 (1−6752T+p5T2)2 ( 1 - 6752 T + p^{5} T^{2} )^{2}
37C22C_2^2 1+56462470T2+p10T4 1 + 56462470 T^{2} + p^{10} T^{4}
41C2C_2 (1−198T+p5T2)2 ( 1 - 198 T + p^{5} T^{2} )^{2}
43C22C_2^2 1−293842250T2+p10T4 1 - 293842250 T^{2} + p^{10} T^{4}
47C22C_2^2 1−347593490T2+p10T4 1 - 347593490 T^{2} + p^{10} T^{4}
53C22C_2^2 1−802472090T2+p10T4 1 - 802472090 T^{2} + p^{10} T^{4}
59C2C_2 (1−24660T+p5T2)2 ( 1 - 24660 T + p^{5} T^{2} )^{2}
61C2C_2 (1+5698T+p5T2)2 ( 1 + 5698 T + p^{5} T^{2} )^{2}
67C22C_2^2 1−795787610T2+p10T4 1 - 795787610 T^{2} + p^{10} T^{4}
71C2C_2 (1+53352T+p5T2)2 ( 1 + 53352 T + p^{5} T^{2} )^{2}
73C22C_2^2 1+883886830T2+p10T4 1 + 883886830 T^{2} + p^{10} T^{4}
79C2C_2 (1−51920T+p5T2)2 ( 1 - 51920 T + p^{5} T^{2} )^{2}
83C22C_2^2 1−4053674810T2+p10T4 1 - 4053674810 T^{2} + p^{10} T^{4}
89C2C_2 (1−9990T+p5T2)2 ( 1 - 9990 T + p^{5} T^{2} )^{2}
97C22C_2^2 1−6923133890T2+p10T4 1 - 6923133890 T^{2} + p^{10} 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

−15.37328280401321787132936946656, −14.32495061373711937131605361349, −13.88533960379769354094494703085, −13.44870478713871244123565871646, −12.98414098550329076800317843724, −12.07099147048482634046765730285, −11.73104783609597120170388968097, −10.57114513803079156106554651776, −10.34247707015070465266509454029, −9.964282482124206858491416597958, −8.911885214124723887739297804694, −8.418048433651762159224634439680, −7.48808201075511211134016280125, −6.44244651590754007683773395272, −6.31654591453785310420190729381, −5.22676072222744257576886987393, −4.57444132457676853726832937226, −2.64897320938473199539532211455, −2.49090330731543460257831706405, −1.02087537862008371301362993346, 1.02087537862008371301362993346, 2.49090330731543460257831706405, 2.64897320938473199539532211455, 4.57444132457676853726832937226, 5.22676072222744257576886987393, 6.31654591453785310420190729381, 6.44244651590754007683773395272, 7.48808201075511211134016280125, 8.418048433651762159224634439680, 8.911885214124723887739297804694, 9.964282482124206858491416597958, 10.34247707015070465266509454029, 10.57114513803079156106554651776, 11.73104783609597120170388968097, 12.07099147048482634046765730285, 12.98414098550329076800317843724, 13.44870478713871244123565871646, 13.88533960379769354094494703085, 14.32495061373711937131605361349, 15.37328280401321787132936946656

Graph of the ZZ-function along the critical line