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Properties

Label 4-95e4-1.1-c1e2-0-5
Degree 44
Conductor 8145062581450625
Sign 11
Analytic cond. 5193.365193.36
Root an. cond. 8.489108.48910
Motivic weight 11
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  + 3·2-s + 4·4-s + 4·7-s + 3·8-s − 9-s + 2·11-s − 4·13-s + 12·14-s + 3·16-s + 3·17-s − 3·18-s + 6·22-s − 2·23-s − 12·26-s + 16·28-s − 12·29-s − 3·31-s + 6·32-s + 9·34-s − 4·36-s + 15·37-s + 15·41-s − 43-s + 8·44-s − 6·46-s + 13·47-s + 3·49-s + ⋯
L(s)  = 1  + 2.12·2-s + 2·4-s + 1.51·7-s + 1.06·8-s − 1/3·9-s + 0.603·11-s − 1.10·13-s + 3.20·14-s + 3/4·16-s + 0.727·17-s − 0.707·18-s + 1.27·22-s − 0.417·23-s − 2.35·26-s + 3.02·28-s − 2.22·29-s − 0.538·31-s + 1.06·32-s + 1.54·34-s − 2/3·36-s + 2.46·37-s + 2.34·41-s − 0.152·43-s + 1.20·44-s − 0.884·46-s + 1.89·47-s + 3/7·49-s + ⋯

Functional equation

Λ(s)=(81450625s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(81450625s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8145062581450625    =    54⋅1945^{4} \cdot 19^{4}
Sign: 11
Analytic conductor: 5193.365193.36
Root analytic conductor: 8.489108.48910
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 81450625, ( :1/2,1/2), 1)(4,\ 81450625,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 13.3694173713.36941737
L(12)L(\frac12) ≈\approx 13.3694173713.36941737
L(32)L(\frac{3}{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)
bad5 1 1
19 1 1
good2C22C_2^2 1−3T+5T2−3pT3+p2T4 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4}
3C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
7D4D_{4} 1−4T+13T2−4pT3+p2T4 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4}
11D4D_{4} 1−2T+3T2−2pT3+p2T4 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
17D4D_{4} 1−3T+25T2−3pT3+p2T4 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+2T+42T2+2pT3+p2T4 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4}
29C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
31D4D_{4} 1+3T+53T2+3pT3+p2T4 1 + 3 T + 53 T^{2} + 3 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−15T+119T2−15pT3+p2T4 1 - 15 T + 119 T^{2} - 15 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−15T+137T2−15pT3+p2T4 1 - 15 T + 137 T^{2} - 15 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+T−15T2+pT3+p2T4 1 + T - 15 T^{2} + p T^{3} + p^{2} T^{4}
47D4D_{4} 1−13T+135T2−13pT3+p2T4 1 - 13 T + 135 T^{2} - 13 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+15T+161T2+15pT3+p2T4 1 + 15 T + 161 T^{2} + 15 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−10T+138T2−10pT3+p2T4 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−9T+41T2−9pT3+p2T4 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+4T+93T2+4pT3+p2T4 1 + 4 T + 93 T^{2} + 4 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−18T+203T2−18pT3+p2T4 1 - 18 T + 203 T^{2} - 18 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−8T+117T2−8pT3+p2T4 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4}
79C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
83D4D_{4} 1−9T+155T2−9pT3+p2T4 1 - 9 T + 155 T^{2} - 9 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−6T+142T2−6pT3+p2T4 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−17T+265T2−17pT3+p2T4 1 - 17 T + 265 T^{2} - 17 p T^{3} + p^{2} 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

−7.70946533386677580161801671705, −7.53199758408307633699694527820, −7.42544825868587483579585807805, −6.65800630512978560427129183879, −6.23808047807277525242572979983, −6.15172373135778251105009087537, −5.56275845217199680373185068814, −5.36523198740411731674930401879, −5.15533855330473720133972005760, −4.86032696566018611235211145461, −4.33576223688831134920011937676, −4.03933060691991415349575302132, −3.91745396200519847767147674306, −3.57101228384936875797749352760, −2.85071817820189945034282567084, −2.58267636702775333577416494771, −2.18123886133608093418840986860, −1.73361146595770377618383141504, −1.13781103842675887960957823458, −0.58815393720946770600808150623, 0.58815393720946770600808150623, 1.13781103842675887960957823458, 1.73361146595770377618383141504, 2.18123886133608093418840986860, 2.58267636702775333577416494771, 2.85071817820189945034282567084, 3.57101228384936875797749352760, 3.91745396200519847767147674306, 4.03933060691991415349575302132, 4.33576223688831134920011937676, 4.86032696566018611235211145461, 5.15533855330473720133972005760, 5.36523198740411731674930401879, 5.56275845217199680373185068814, 6.15172373135778251105009087537, 6.23808047807277525242572979983, 6.65800630512978560427129183879, 7.42544825868587483579585807805, 7.53199758408307633699694527820, 7.70946533386677580161801671705

Graph of the ZZ-function along the critical line