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Properties

Label 4-24e2-1.1-c2e2-0-0
Degree 44
Conductor 576576
Sign 11
Analytic cond. 0.4276540.427654
Root an. cond. 0.8086730.808673
Motivic weight 22
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  + 2·3-s − 12·7-s − 5·9-s + 20·13-s + 4·19-s − 24·21-s + 18·25-s − 28·27-s − 44·31-s − 12·37-s + 40·39-s + 164·43-s + 10·49-s + 8·57-s − 172·61-s + 60·63-s + 4·67-s + 164·73-s + 36·75-s + 20·79-s − 11·81-s − 240·91-s − 88·93-s − 188·97-s − 268·103-s + 20·109-s − 24·111-s + ⋯
L(s)  = 1  + 2/3·3-s − 1.71·7-s − 5/9·9-s + 1.53·13-s + 4/19·19-s − 8/7·21-s + 0.719·25-s − 1.03·27-s − 1.41·31-s − 0.324·37-s + 1.02·39-s + 3.81·43-s + 0.204·49-s + 8/57·57-s − 2.81·61-s + 0.952·63-s + 4/67·67-s + 2.24·73-s + 0.479·75-s + 0.253·79-s − 0.135·81-s − 2.63·91-s − 0.946·93-s − 1.93·97-s − 2.60·103-s + 0.183·109-s − 0.216·111-s + ⋯

Functional equation

Λ(s)=(576s/2 ΓC(s)2 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(576s/2 ΓC(s+1)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 576576    =    26⋅322^{6} \cdot 3^{2}
Sign: 11
Analytic conductor: 0.4276540.427654
Root analytic conductor: 0.8086730.808673
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 576, ( :1,1), 1)(4,\ 576,\ (\ :1, 1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.88827507020.8882750702
L(12)L(\frac12) ≈\approx 0.88827507020.8882750702
L(2)L(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
3C2C_2 1−2T+p2T2 1 - 2 T + p^{2} T^{2}
good5C22C_2^2 1−18T2+p4T4 1 - 18 T^{2} + p^{4} T^{4}
7C2C_2 (1+6T+p2T2)2 ( 1 + 6 T + p^{2} T^{2} )^{2}
11C22C_2^2 1−210T2+p4T4 1 - 210 T^{2} + p^{4} T^{4}
13C2C_2 (1−10T+p2T2)2 ( 1 - 10 T + p^{2} T^{2} )^{2}
17C22C_2^2 1−66T2+p4T4 1 - 66 T^{2} + p^{4} T^{4}
19C2C_2 (1−2T+p2T2)2 ( 1 - 2 T + p^{2} T^{2} )^{2}
23C22C_2^2 1−930T2+p4T4 1 - 930 T^{2} + p^{4} T^{4}
29C22C_2^2 1−1394T2+p4T4 1 - 1394 T^{2} + p^{4} T^{4}
31C2C_2 (1+22T+p2T2)2 ( 1 + 22 T + p^{2} T^{2} )^{2}
37C2C_2 (1+6T+p2T2)2 ( 1 + 6 T + p^{2} T^{2} )^{2}
41C22C_2^2 1−2210T2+p4T4 1 - 2210 T^{2} + p^{4} T^{4}
43C2C_2 (1−82T+p2T2)2 ( 1 - 82 T + p^{2} T^{2} )^{2}
47C22C_2^2 1+190T2+p4T4 1 + 190 T^{2} + p^{4} T^{4}
53C22C_2^2 1−1746T2+p4T4 1 - 1746 T^{2} + p^{4} T^{4}
59C22C_2^2 1−1554T2+p4T4 1 - 1554 T^{2} + p^{4} T^{4}
61C2C_2 (1+86T+p2T2)2 ( 1 + 86 T + p^{2} T^{2} )^{2}
67C2C_2 (1−2T+p2T2)2 ( 1 - 2 T + p^{2} T^{2} )^{2}
71C22C_2^2 1+5406T2+p4T4 1 + 5406 T^{2} + p^{4} T^{4}
73C2C_2 (1−82T+p2T2)2 ( 1 - 82 T + p^{2} T^{2} )^{2}
79C2C_2 (1−10T+p2T2)2 ( 1 - 10 T + p^{2} T^{2} )^{2}
83C22C_2^2 1−8370T2+p4T4 1 - 8370 T^{2} + p^{4} T^{4}
89C22C_2^2 1−14690T2+p4T4 1 - 14690 T^{2} + p^{4} T^{4}
97C2C_2 (1+94T+p2T2)2 ( 1 + 94 T + p^{2} 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

−17.93329962250395857674990055759, −17.14668355749950927793260670781, −16.28425281770056576603944690490, −16.16509422432547218507615041236, −15.41817941221427843923117773211, −14.73168390092593549178382905593, −13.77100009717908332975017242392, −13.70006705163429983926489973324, −12.61782911242619031988447152576, −12.46828269382948081457220826153, −10.99342490422542368991926475278, −10.84205192665977700251879871973, −9.396373175556693518577016958289, −9.311841927145405240809802494724, −8.396629698352459685727855168211, −7.40713148646346123493409827011, −6.36573606719747848689709829916, −5.72798389202971793650888671793, −3.84755496424067939340094305126, −2.98246483718788002904300333538, 2.98246483718788002904300333538, 3.84755496424067939340094305126, 5.72798389202971793650888671793, 6.36573606719747848689709829916, 7.40713148646346123493409827011, 8.396629698352459685727855168211, 9.311841927145405240809802494724, 9.396373175556693518577016958289, 10.84205192665977700251879871973, 10.99342490422542368991926475278, 12.46828269382948081457220826153, 12.61782911242619031988447152576, 13.70006705163429983926489973324, 13.77100009717908332975017242392, 14.73168390092593549178382905593, 15.41817941221427843923117773211, 16.16509422432547218507615041236, 16.28425281770056576603944690490, 17.14668355749950927793260670781, 17.93329962250395857674990055759

Graph of the ZZ-function along the critical line