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Properties

Label 4-1110e2-1.1-c3e2-0-1
Degree 44
Conductor 12321001232100
Sign 11
Analytic cond. 4289.214289.21
Root an. cond. 8.092728.09272
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 6·3-s + 12·4-s + 10·5-s + 24·6-s − 30·7-s + 32·8-s + 27·9-s + 40·10-s − 77·11-s + 72·12-s − 36·13-s − 120·14-s + 60·15-s + 80·16-s − 65·17-s + 108·18-s − 55·19-s + 120·20-s − 180·21-s − 308·22-s − 128·23-s + 192·24-s + 75·25-s − 144·26-s + 108·27-s − 360·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 1.61·7-s + 1.41·8-s + 9-s + 1.26·10-s − 2.11·11-s + 1.73·12-s − 0.768·13-s − 2.29·14-s + 1.03·15-s + 5/4·16-s − 0.927·17-s + 1.41·18-s − 0.664·19-s + 1.34·20-s − 1.87·21-s − 2.98·22-s − 1.16·23-s + 1.63·24-s + 3/5·25-s − 1.08·26-s + 0.769·27-s − 2.42·28-s + ⋯

Functional equation

Λ(s)=(1232100s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1232100s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12321001232100    =    22⋅32⋅52⋅3722^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}
Sign: 11
Analytic conductor: 4289.214289.21
Root analytic conductor: 8.092728.09272
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 1232100, ( :3/2,3/2), 1)(4,\ 1232100,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad2C1C_1 (1−pT)2 ( 1 - p T )^{2}
3C1C_1 (1−pT)2 ( 1 - p T )^{2}
5C1C_1 (1−pT)2 ( 1 - p T )^{2}
37C1C_1 (1+pT)2 ( 1 + p T )^{2}
good7C2C_2 (1+15T+p3T2)2 ( 1 + 15 T + p^{3} T^{2} )^{2}
11D4D_{4} 1+7pT+4007T2+7p4T3+p6T4 1 + 7 p T + 4007 T^{2} + 7 p^{4} T^{3} + p^{6} T^{4}
13D4D_{4} 1+36T+3193T2+36p3T3+p6T4 1 + 36 T + 3193 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+65T+10501T2+65p3T3+p6T4 1 + 65 T + 10501 T^{2} + 65 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+55T+8969T2+55p3T3+p6T4 1 + 55 T + 8969 T^{2} + 55 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+128T+18121T2+128p3T3+p6T4 1 + 128 T + 18121 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+53T+12865T2+53p3T3+p6T4 1 + 53 T + 12865 T^{2} + 53 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+388T+86909T2+388p3T3+p6T4 1 + 388 T + 86909 T^{2} + 388 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−118T+12247T2−118p3T3+p6T4 1 - 118 T + 12247 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+353T+147329T2+353p3T3+p6T4 1 + 353 T + 147329 T^{2} + 353 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+146T+209071T2+146p3T3+p6T4 1 + 146 T + 209071 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+448T+128330T2+448p3T3+p6T4 1 + 448 T + 128330 T^{2} + 448 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+17T+34933T2+17p3T3+p6T4 1 + 17 T + 34933 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+18T+383527T2+18p3T3+p6T4 1 + 18 T + 383527 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−147T+525661T2−147p3T3+p6T4 1 - 147 T + 525661 T^{2} - 147 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+45T+584431T2+45p3T3+p6T4 1 + 45 T + 584431 T^{2} + 45 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−163T+750989T2−163p3T3+p6T4 1 - 163 T + 750989 T^{2} - 163 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+53T+975663T2+53p3T3+p6T4 1 + 53 T + 975663 T^{2} + 53 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+1389T+1372129T2+1389p3T3+p6T4 1 + 1389 T + 1372129 T^{2} + 1389 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+1018T+1334983T2+1018p3T3+p6T4 1 + 1018 T + 1334983 T^{2} + 1018 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−2318T+3080543T2−2318p3T3+p6T4 1 - 2318 T + 3080543 T^{2} - 2318 p^{3} T^{3} + p^{6} 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

−9.208502415486305368825616976892, −9.067390319827868482280830045114, −8.215241971356809728310252408823, −8.082150181684514821760595250866, −7.31401712948829052649502822530, −7.27534755602640002920595316717, −6.49911929555020335862029939160, −6.43016238123071192543092571516, −5.64883855075278718172805936018, −5.54158566327720649219651126236, −4.73061997157057649994681705515, −4.65056697063125303034925956830, −3.62312777389315297434712970495, −3.57773536804614854620816769602, −2.74163251680221382164206737512, −2.71833802875145940837054996106, −1.88738428039713629892708528973, −1.88138243943507535629368002119, 0, 0, 1.88138243943507535629368002119, 1.88738428039713629892708528973, 2.71833802875145940837054996106, 2.74163251680221382164206737512, 3.57773536804614854620816769602, 3.62312777389315297434712970495, 4.65056697063125303034925956830, 4.73061997157057649994681705515, 5.54158566327720649219651126236, 5.64883855075278718172805936018, 6.43016238123071192543092571516, 6.49911929555020335862029939160, 7.27534755602640002920595316717, 7.31401712948829052649502822530, 8.082150181684514821760595250866, 8.215241971356809728310252408823, 9.067390319827868482280830045114, 9.208502415486305368825616976892

Graph of the ZZ-function along the critical line