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Properties

Label 4-2736e2-1.1-c1e2-0-39
Degree 44
Conductor 74856967485696
Sign 11
Analytic cond. 477.294477.294
Root an. cond. 4.674084.67408
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 3·7-s + 3·11-s + 12·13-s − 17-s − 2·19-s + 8·23-s + 25-s − 4·29-s − 6·31-s + 3·35-s + 6·37-s − 6·41-s − 43-s + 5·47-s + 3·49-s + 12·53-s + 3·55-s − 8·59-s + 9·61-s + 12·65-s + 24·67-s + 7·73-s + 9·77-s − 4·79-s + 4·83-s − 85-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.13·7-s + 0.904·11-s + 3.32·13-s − 0.242·17-s − 0.458·19-s + 1.66·23-s + 1/5·25-s − 0.742·29-s − 1.07·31-s + 0.507·35-s + 0.986·37-s − 0.937·41-s − 0.152·43-s + 0.729·47-s + 3/7·49-s + 1.64·53-s + 0.404·55-s − 1.04·59-s + 1.15·61-s + 1.48·65-s + 2.93·67-s + 0.819·73-s + 1.02·77-s − 0.450·79-s + 0.439·83-s − 0.108·85-s + ⋯

Functional equation

Λ(s)=(7485696s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7485696s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 74856967485696    =    28⋅34⋅1922^{8} \cdot 3^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 477.294477.294
Root analytic conductor: 4.674084.67408
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 7485696, ( :1/2,1/2), 1)(4,\ 7485696,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.4803994935.480399493
L(12)L(\frac12) ≈\approx 5.4803994935.480399493
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)
bad2 1 1
3 1 1
19C1C_1 (1+T)2 ( 1 + T )^{2}
good5D4D_{4} 1−T−pT3+p2T4 1 - T - p T^{3} + p^{2} T^{4}
7D4D_{4} 1−3T+6T2−3pT3+p2T4 1 - 3 T + 6 T^{2} - 3 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−3T+14T2−3pT3+p2T4 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4}
13C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
17D4D_{4} 1+T+24T2+pT3+p2T4 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4}
23C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31D4D_{4} 1+6T+30T2+6pT3+p2T4 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−6T+42T2−6pT3+p2T4 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+6T+50T2+6pT3+p2T4 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+T−6T2+pT3+p2T4 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4}
47D4D_{4} 1−5T+90T2−5pT3+p2T4 1 - 5 T + 90 T^{2} - 5 p T^{3} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
61D4D_{4} 1−9T+132T2−9pT3+p2T4 1 - 9 T + 132 T^{2} - 9 p T^{3} + p^{2} T^{4}
67C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73D4D_{4} 1−7T+148T2−7pT3+p2T4 1 - 7 T + 148 T^{2} - 7 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+4T−2T2+4pT3+p2T4 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−4T+6T2−4pT3+p2T4 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
97C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
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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

−8.921911727116791371426556241022, −8.631340419532668056649987861946, −8.313422434577610782703747839098, −8.163241199022309603922342786487, −7.43932233232358196747629989360, −7.06630649185132370527665321635, −6.68932255664833789415208962646, −6.33902792531288865110626768653, −5.92766480130418452316482589468, −5.63465522142915149935223728988, −5.11659551098610445376251249123, −4.82933858274108458243299412201, −4.03433540767028760332384776868, −3.83944256472822148577496343454, −3.61821290788363792929502287498, −2.93772816651841095731164005897, −2.15565656473715690413216233405, −1.78932366113614628914885984326, −1.04398047849916943650448216363, −1.02783915716178684131355557050, 1.02783915716178684131355557050, 1.04398047849916943650448216363, 1.78932366113614628914885984326, 2.15565656473715690413216233405, 2.93772816651841095731164005897, 3.61821290788363792929502287498, 3.83944256472822148577496343454, 4.03433540767028760332384776868, 4.82933858274108458243299412201, 5.11659551098610445376251249123, 5.63465522142915149935223728988, 5.92766480130418452316482589468, 6.33902792531288865110626768653, 6.68932255664833789415208962646, 7.06630649185132370527665321635, 7.43932233232358196747629989360, 8.163241199022309603922342786487, 8.313422434577610782703747839098, 8.631340419532668056649987861946, 8.921911727116791371426556241022

Graph of the ZZ-function along the critical line