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Properties

Label 4-9702e2-1.1-c1e2-0-27
Degree 44
Conductor 9412880494128804
Sign 11
Analytic cond. 6001.736001.73
Root an. cond. 8.801758.80175
Motivic weight 11
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  + 2·2-s + 3·4-s + 4·8-s − 2·11-s − 4·13-s + 5·16-s − 8·17-s − 4·19-s − 4·22-s − 8·26-s + 8·29-s − 12·31-s + 6·32-s − 16·34-s + 4·37-s − 8·38-s + 8·41-s − 6·44-s − 4·47-s − 12·52-s − 4·53-s + 16·58-s + 16·59-s − 20·61-s − 24·62-s + 7·64-s − 4·67-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s − 0.603·11-s − 1.10·13-s + 5/4·16-s − 1.94·17-s − 0.917·19-s − 0.852·22-s − 1.56·26-s + 1.48·29-s − 2.15·31-s + 1.06·32-s − 2.74·34-s + 0.657·37-s − 1.29·38-s + 1.24·41-s − 0.904·44-s − 0.583·47-s − 1.66·52-s − 0.549·53-s + 2.10·58-s + 2.08·59-s − 2.56·61-s − 3.04·62-s + 7/8·64-s − 0.488·67-s + ⋯

Functional equation

Λ(s)=(94128804s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(94128804s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9412880494128804    =    22⋅34⋅74⋅1122^{2} \cdot 3^{4} \cdot 7^{4} \cdot 11^{2}
Sign: 11
Analytic conductor: 6001.736001.73
Root analytic conductor: 8.801758.80175
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 94128804, ( :1/2,1/2), 1)(4,\ 94128804,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad2C1C_1 (1−T)2 ( 1 - T )^{2}
3 1 1
7 1 1
11C1C_1 (1+T)2 ( 1 + T )^{2}
good5C22C_2^2 1+p2T4 1 + p^{2} T^{4}
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
17D4D_{4} 1+8T+40T2+8pT3+p2T4 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4}
19D4D_{4} 1+4T+32T2+4pT3+p2T4 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
29C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
31D4D_{4} 1+12T+88T2+12pT3+p2T4 1 + 12 T + 88 T^{2} + 12 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−4T+38T2−4pT3+p2T4 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−8T+88T2−8pT3+p2T4 1 - 8 T + 88 T^{2} - 8 p T^{3} + p^{2} T^{4}
43C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
47C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+4T+70T2+4pT3+p2T4 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−16T+142T2−16pT3+p2T4 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67D4D_{4} 1+4T+98T2+4pT3+p2T4 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4}
71C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
73D4D_{4} 1+8T+72T2+8pT3+p2T4 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−4T+122T2−4pT3+p2T4 1 - 4 T + 122 T^{2} - 4 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−4T+160T2−4pT3+p2T4 1 - 4 T + 160 T^{2} - 4 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+20T+238T2+20pT3+p2T4 1 + 20 T + 238 T^{2} + 20 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+4T+38T2+4pT3+p2T4 1 + 4 T + 38 T^{2} + 4 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

−7.40390696552429875805816987607, −7.07149514668228748995453113511, −6.78633192065727996855660760536, −6.40619557849196721234626637624, −6.05192671395370405034974781964, −5.88568236290735424417634580988, −5.30568208992729320133223596045, −5.07365129834296019028673126943, −4.57417664900218053085768901530, −4.55106686279006346469585476986, −4.03230845911111815865991516512, −3.87676902674504075288729074028, −3.13243545207358721453027083435, −2.80952576827936694236965068411, −2.51016169042402406257076724758, −2.20547063448227657421133644363, −1.71368723360904622860530757896, −1.19238049380801914993518122461, 0, 0, 1.19238049380801914993518122461, 1.71368723360904622860530757896, 2.20547063448227657421133644363, 2.51016169042402406257076724758, 2.80952576827936694236965068411, 3.13243545207358721453027083435, 3.87676902674504075288729074028, 4.03230845911111815865991516512, 4.55106686279006346469585476986, 4.57417664900218053085768901530, 5.07365129834296019028673126943, 5.30568208992729320133223596045, 5.88568236290735424417634580988, 6.05192671395370405034974781964, 6.40619557849196721234626637624, 6.78633192065727996855660760536, 7.07149514668228748995453113511, 7.40390696552429875805816987607

Graph of the ZZ-function along the critical line