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Properties

Label 4-45e2-1.1-c5e2-0-3
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  + 5·2-s − 9·4-s + 50·5-s + 80·7-s − 55·8-s + 250·10-s + 800·11-s − 120·13-s + 400·14-s − 193·16-s + 1.94e3·17-s − 1.51e3·19-s − 450·20-s + 4.00e3·22-s − 1.32e3·23-s + 1.87e3·25-s − 600·26-s − 720·28-s + 1.30e3·29-s − 5.82e3·31-s − 5.65e3·32-s + 9.70e3·34-s + 4.00e3·35-s − 1.25e4·37-s − 7.56e3·38-s − 2.75e3·40-s + 400·41-s + ⋯
L(s)  = 1  + 0.883·2-s − 0.281·4-s + 0.894·5-s + 0.617·7-s − 0.303·8-s + 0.790·10-s + 1.99·11-s − 0.196·13-s + 0.545·14-s − 0.188·16-s + 1.62·17-s − 0.960·19-s − 0.251·20-s + 1.76·22-s − 0.520·23-s + 3/5·25-s − 0.174·26-s − 0.173·28-s + 0.287·29-s − 1.08·31-s − 0.976·32-s + 1.43·34-s + 0.551·35-s − 1.50·37-s − 0.849·38-s − 0.271·40-s + 0.0371·41-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 4.3130093694.313009369
L(12)L(\frac12) ≈\approx 4.3130093694.313009369
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
5C1C_1 (1−p2T)2 ( 1 - p^{2} T )^{2}
good2D4D_{4} 1−5T+17pT2−5p5T3+p10T4 1 - 5 T + 17 p T^{2} - 5 p^{5} T^{3} + p^{10} T^{4}
7D4D_{4} 1−80T+20714T2−80p5T3+p10T4 1 - 80 T + 20714 T^{2} - 80 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 1−800T+467602T2−800p5T3+p10T4 1 - 800 T + 467602 T^{2} - 800 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1+120T+514186T2+120p5T3+p10T4 1 + 120 T + 514186 T^{2} + 120 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 1−1940T+3743494T2−1940p5T3+p10T4 1 - 1940 T + 3743494 T^{2} - 1940 p^{5} T^{3} + p^{10} T^{4}
19D4D_{4} 1+1512T+1811734T2+1512p5T3+p10T4 1 + 1512 T + 1811734 T^{2} + 1512 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 1+1320T+4100206T2+1320p5T3+p10T4 1 + 1320 T + 4100206 T^{2} + 1320 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1−1300T−17947202T2−1300p5T3+p10T4 1 - 1300 T - 17947202 T^{2} - 1300 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 1+5824T+58720046T2+5824p5T3+p10T4 1 + 5824 T + 58720046 T^{2} + 5824 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 1+12560T+102958314T2+12560p5T3+p10T4 1 + 12560 T + 102958314 T^{2} + 12560 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 1−400T+164704402T2−400p5T3+p10T4 1 - 400 T + 164704402 T^{2} - 400 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1−25680T+399490486T2−25680p5T3+p10T4 1 - 25680 T + 399490486 T^{2} - 25680 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1−18920T+546954334T2−18920p5T3+p10T4 1 - 18920 T + 546954334 T^{2} - 18920 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 1−49460T+1434563566T2−49460p5T3+p10T4 1 - 49460 T + 1434563566 T^{2} - 49460 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1−63200T+2393594098T2−63200p5T3+p10T4 1 - 63200 T + 2393594098 T^{2} - 63200 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 1+49116T+1219519966T2+49116p5T3+p10T4 1 + 49116 T + 1219519966 T^{2} + 49116 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 1−6080T+319369814T2−6080p5T3+p10T4 1 - 6080 T + 319369814 T^{2} - 6080 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1−65200T+4591816702T2−65200p5T3+p10T4 1 - 65200 T + 4591816702 T^{2} - 65200 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 1−97740T+5753972086T2−97740p5T3+p10T4 1 - 97740 T + 5753972086 T^{2} - 97740 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 1+46288T+6429395534T2+46288p5T3+p10T4 1 + 46288 T + 6429395534 T^{2} + 46288 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 1+57360T+4528611766T2+57360p5T3+p10T4 1 + 57360 T + 4528611766 T^{2} + 57360 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1+87000T+10502568898T2+87000p5T3+p10T4 1 + 87000 T + 10502568898 T^{2} + 87000 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1−10180T+16899916614T2−10180p5T3+p10T4 1 - 10180 T + 16899916614 T^{2} - 10180 p^{5} T^{3} + 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

−14.66784970969022019510655850936, −14.37784536888453374327419702333, −14.03683634096722307094210643198, −13.68496200415498815155107445593, −12.61180369256250028057802710003, −12.48918967488013551504117405756, −11.77617916094901859396700299413, −11.01034771856315258632575862221, −10.32377565394036678169206692921, −9.596994509059774957244181837257, −9.023182964180726345828141504177, −8.460563802580778649482788226914, −7.35453074034000674602867122277, −6.66667448731695673263077309964, −5.70691639151721770956250193902, −5.29318868746665421033732889688, −4.15975404841838252730105105759, −3.76406847311085762487158546967, −2.13598064014564555158892296757, −1.11486975673290849705631102973, 1.11486975673290849705631102973, 2.13598064014564555158892296757, 3.76406847311085762487158546967, 4.15975404841838252730105105759, 5.29318868746665421033732889688, 5.70691639151721770956250193902, 6.66667448731695673263077309964, 7.35453074034000674602867122277, 8.460563802580778649482788226914, 9.023182964180726345828141504177, 9.596994509059774957244181837257, 10.32377565394036678169206692921, 11.01034771856315258632575862221, 11.77617916094901859396700299413, 12.48918967488013551504117405756, 12.61180369256250028057802710003, 13.68496200415498815155107445593, 14.03683634096722307094210643198, 14.37784536888453374327419702333, 14.66784970969022019510655850936

Graph of the ZZ-function along the critical line