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Properties

Label 4-8550e2-1.1-c1e2-0-7
Degree 44
Conductor 7310250073102500
Sign 11
Analytic cond. 4661.074661.07
Root an. cond. 8.262698.26269
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  + 2·2-s + 3·4-s + 6·7-s + 4·8-s + 6·13-s + 12·14-s + 5·16-s − 2·17-s − 2·19-s − 10·23-s + 12·26-s + 18·28-s + 6·29-s + 4·31-s + 6·32-s − 4·34-s − 4·38-s + 12·43-s − 20·46-s + 15·49-s + 18·52-s + 6·53-s + 24·56-s + 12·58-s + 6·59-s + 20·61-s + 8·62-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 2.26·7-s + 1.41·8-s + 1.66·13-s + 3.20·14-s + 5/4·16-s − 0.485·17-s − 0.458·19-s − 2.08·23-s + 2.35·26-s + 3.40·28-s + 1.11·29-s + 0.718·31-s + 1.06·32-s − 0.685·34-s − 0.648·38-s + 1.82·43-s − 2.94·46-s + 15/7·49-s + 2.49·52-s + 0.824·53-s + 3.20·56-s + 1.57·58-s + 0.781·59-s + 2.56·61-s + 1.01·62-s + ⋯

Functional equation

Λ(s)=(73102500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(73102500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 7310250073102500    =    22⋅34⋅54⋅1922^{2} \cdot 3^{4} \cdot 5^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 4661.074661.07
Root analytic conductor: 8.262698.26269
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 73102500, ( :1/2,1/2), 1)(4,\ 73102500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 16.9728487216.97284872
L(12)L(\frac12) ≈\approx 16.9728487216.97284872
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
5 1 1
19C1C_1 (1+T)2 ( 1 + T )^{2}
good7D4D_{4} 1−6T+3pT2−6pT3+p2T4 1 - 6 T + 3 p T^{2} - 6 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+20T2+p2T4 1 + 20 T^{2} + p^{2} T^{4}
13D4D_{4} 1−6T+27T2−6pT3+p2T4 1 - 6 T + 27 T^{2} - 6 p T^{3} + p^{2} T^{4}
17C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
23D4D_{4} 1+10T+53T2+10pT3+p2T4 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4}
29D4D_{4} 1−6T+59T2−6pT3+p2T4 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−4T+48T2−4pT3+p2T4 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
41C22C_2^2 1+64T2+p2T4 1 + 64 T^{2} + p^{2} T^{4}
43D4D_{4} 1−12T+104T2−12pT3+p2T4 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53D4D_{4} 1−6T+43T2−6pT3+p2T4 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−6T+29T2−6pT3+p2T4 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−20T+204T2−20pT3+p2T4 1 - 20 T + 204 T^{2} - 20 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−18T+197T2−18pT3+p2T4 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−24T+4pT2−24pT3+p2T4 1 - 24 T + 4 p T^{2} - 24 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−6T+83T2−6pT3+p2T4 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4}
79C4C_4 1−4T+90T2−4pT3+p2T4 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+12T+130T2+12pT3+p2T4 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+128T2+p2T4 1 + 128 T^{2} + p^{2} T^{4}
97D4D_{4} 1+12T+198T2+12pT3+p2T4 1 + 12 T + 198 T^{2} + 12 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

−7.916198572879353642813300380482, −7.908348515390718593603931495463, −6.97283679067666865370931078098, −6.90887708408965047979866863337, −6.38626063195253612822281471128, −6.31701654619386153804348470004, −5.62964146597898368826261347087, −5.55814047309776613225682615586, −5.01576708376672632726759356209, −5.00485273216513622461034039567, −4.20467668668607205994053930669, −4.17580433616505669081611166662, −3.78585933314799342505132112432, −3.72324330251560670835138603376, −2.65624352072039710110842186887, −2.51612897798450994033934438120, −2.08197217236639860443911343436, −1.72455973947737535497831794449, −1.10142381880238312606238711056, −0.803923554419488282220457854311, 0.803923554419488282220457854311, 1.10142381880238312606238711056, 1.72455973947737535497831794449, 2.08197217236639860443911343436, 2.51612897798450994033934438120, 2.65624352072039710110842186887, 3.72324330251560670835138603376, 3.78585933314799342505132112432, 4.17580433616505669081611166662, 4.20467668668607205994053930669, 5.00485273216513622461034039567, 5.01576708376672632726759356209, 5.55814047309776613225682615586, 5.62964146597898368826261347087, 6.31701654619386153804348470004, 6.38626063195253612822281471128, 6.90887708408965047979866863337, 6.97283679067666865370931078098, 7.908348515390718593603931495463, 7.916198572879353642813300380482

Graph of the ZZ-function along the critical line