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Properties

Label 4-9702e2-1.1-c1e2-0-35
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 + 2·5-s + 4·8-s + 4·10-s + 2·11-s − 8·13-s + 5·16-s − 10·17-s − 4·19-s + 6·20-s + 4·22-s + 2·23-s − 5·25-s − 16·26-s + 4·31-s + 6·32-s − 20·34-s + 8·37-s − 8·38-s + 8·40-s − 2·41-s + 6·44-s + 4·46-s − 10·47-s − 10·50-s − 24·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.41·8-s + 1.26·10-s + 0.603·11-s − 2.21·13-s + 5/4·16-s − 2.42·17-s − 0.917·19-s + 1.34·20-s + 0.852·22-s + 0.417·23-s − 25-s − 3.13·26-s + 0.718·31-s + 1.06·32-s − 3.42·34-s + 1.31·37-s − 1.29·38-s + 1.26·40-s − 0.312·41-s + 0.904·44-s + 0.589·46-s − 1.45·47-s − 1.41·50-s − 3.32·52-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}
good5D4D_{4} 1−2T+9T2−2pT3+p2T4 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4}
13D4D_{4} 1+8T+34T2+8pT3+p2T4 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+10T+3pT2+10pT3+p2T4 1 + 10 T + 3 p T^{2} + 10 p T^{3} + p^{2} T^{4}
19D4D_{4} 1+4T+34T2+4pT3+p2T4 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4}
23D4D_{4} 1−2T+29T2−2pT3+p2T4 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
31C4C_4 1−4T−6T2−4pT3+p2T4 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−8T+58T2−8pT3+p2T4 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+2T+51T2+2pT3+p2T4 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4}
43C22C_2^2 1+78T2+p2T4 1 + 78 T^{2} + p^{2} T^{4}
47D4D_{4} 1+10T+101T2+10pT3+p2T4 1 + 10 T + 101 T^{2} + 10 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+16T+162T2+16pT3+p2T4 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−4T+90T2−4pT3+p2T4 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+6T+129T2+6pT3+p2T4 1 + 6 T + 129 T^{2} + 6 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+10T+87T2+10pT3+p2T4 1 + 10 T + 87 T^{2} + 10 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−4T+18T2−4pT3+p2T4 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+4T+142T2+4pT3+p2T4 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−18T+221T2−18pT3+p2T4 1 - 18 T + 221 T^{2} - 18 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+14T+207T2+14pT3+p2T4 1 + 14 T + 207 T^{2} + 14 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−8T+122T2−8pT3+p2T4 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+26T+355T2+26pT3+p2T4 1 + 26 T + 355 T^{2} + 26 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

−7.32999472949357098542493411408, −6.93914546361762739765234555412, −6.66783281430768160092629832549, −6.48248190684088067162223539998, −6.03380219116260970618320059631, −6.02072372443107454468346866447, −5.23177298879784910581648850926, −5.04873968736802034067633114709, −4.76325978676030346993989445654, −4.46621911429606557408742049254, −4.00648231924344030377544372363, −3.94628670646552474249604495882, −3.11034096175556012318085943646, −2.72589083661860230681705531101, −2.39925681400888753539627544740, −2.29112392372240368614616774625, −1.58788065413354132868664297900, −1.43008990398391180455392037500, 0, 0, 1.43008990398391180455392037500, 1.58788065413354132868664297900, 2.29112392372240368614616774625, 2.39925681400888753539627544740, 2.72589083661860230681705531101, 3.11034096175556012318085943646, 3.94628670646552474249604495882, 4.00648231924344030377544372363, 4.46621911429606557408742049254, 4.76325978676030346993989445654, 5.04873968736802034067633114709, 5.23177298879784910581648850926, 6.02072372443107454468346866447, 6.03380219116260970618320059631, 6.48248190684088067162223539998, 6.66783281430768160092629832549, 6.93914546361762739765234555412, 7.32999472949357098542493411408

Graph of the ZZ-function along the critical line