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Properties

Label 4-288e2-1.1-c3e2-0-0
Degree 44
Conductor 8294482944
Sign 11
Analytic cond. 288.746288.746
Root an. cond. 4.122204.12220
Motivic weight 33
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  + 16·7-s + 28·17-s − 304·23-s + 138·25-s − 448·31-s + 140·41-s + 672·47-s − 494·49-s − 144·71-s − 588·73-s + 928·79-s − 532·89-s + 1.98e3·97-s − 2.35e3·103-s + 3.42e3·113-s + 448·119-s + 2.41e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4.86e3·161-s + 163-s + 167-s + ⋯
L(s)  = 1  + 0.863·7-s + 0.399·17-s − 2.75·23-s + 1.10·25-s − 2.59·31-s + 0.533·41-s + 2.08·47-s − 1.44·49-s − 0.240·71-s − 0.942·73-s + 1.32·79-s − 0.633·89-s + 2.08·97-s − 2.24·103-s + 2.84·113-s + 0.345·119-s + 1.81·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s − 2.38·161-s + 0.000480·163-s + 0.000463·167-s + ⋯

Functional equation

Λ(s)=(82944s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(82944s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8294482944    =    210⋅342^{10} \cdot 3^{4}
Sign: 11
Analytic conductor: 288.746288.746
Root analytic conductor: 4.122204.12220
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 82944, ( :3/2,3/2), 1)(4,\ 82944,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.0874811352.087481135
L(12)L(\frac12) ≈\approx 2.0874811352.087481135
L(52)L(\frac{5}{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
good5C22C_2^2 1−138T2+p6T4 1 - 138 T^{2} + p^{6} T^{4}
7C2C_2 (1−8T+p3T2)2 ( 1 - 8 T + p^{3} T^{2} )^{2}
11C22C_2^2 1−2410T2+p6T4 1 - 2410 T^{2} + p^{6} T^{4}
13C22C_2^2 1−1594T2+p6T4 1 - 1594 T^{2} + p^{6} T^{4}
17C2C_2 (1−14T+p3T2)2 ( 1 - 14 T + p^{3} T^{2} )^{2}
19C22C_2^2 1−12346T2+p6T4 1 - 12346 T^{2} + p^{6} T^{4}
23C2C_2 (1+152T+p3T2)2 ( 1 + 152 T + p^{3} T^{2} )^{2}
29C22C_2^2 1−23578T2+p6T4 1 - 23578 T^{2} + p^{6} T^{4}
31C2C_2 (1+224T+p3T2)2 ( 1 + 224 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−42058T2+p6T4 1 - 42058 T^{2} + p^{6} T^{4}
41C2C_2 (1−70T+p3T2)2 ( 1 - 70 T + p^{3} T^{2} )^{2}
43C22C_2^2 1+33878T2+p6T4 1 + 33878 T^{2} + p^{6} T^{4}
47C2C_2 (1−336T+p3T2)2 ( 1 - 336 T + p^{3} T^{2} )^{2}
53C22C_2^2 1−296746T2+p6T4 1 - 296746 T^{2} + p^{6} T^{4}
59C22C_2^2 1−125130T2+p6T4 1 - 125130 T^{2} + p^{6} T^{4}
61C22C_2^2 1−444890T2+p6T4 1 - 444890 T^{2} + p^{6} T^{4}
67C22C_2^2 1−571034T2+p6T4 1 - 571034 T^{2} + p^{6} T^{4}
71C2C_2 (1+72T+p3T2)2 ( 1 + 72 T + p^{3} T^{2} )^{2}
73C2C_2 (1+294T+p3T2)2 ( 1 + 294 T + p^{3} T^{2} )^{2}
79C2C_2 (1−464T+p3T2)2 ( 1 - 464 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−846522T2+p6T4 1 - 846522 T^{2} + p^{6} T^{4}
89C2C_2 (1+266T+p3T2)2 ( 1 + 266 T + p^{3} T^{2} )^{2}
97C2C_2 (1−994T+p3T2)2 ( 1 - 994 T + p^{3} 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

−11.50905026458588040193845583089, −11.08809667607036098817194099997, −10.83158076103272159622626065689, −10.11595206648191875139813179680, −9.882038112594304649892676144748, −9.132515518959106249642898717489, −8.799322300174788621536040335678, −8.198969339855747519934614788121, −7.72570470739018124956570362382, −7.39109806090272817479364518718, −6.76704944160135269780785338037, −5.84985551904905678596625356553, −5.80721919713589773528495381596, −5.00570808115154024738561017517, −4.37683703969393919930731561558, −3.84827417093282664065670377297, −3.16123559013333408038601033295, −2.09989410743420151210019375166, −1.71875524449671354734302091813, −0.54128679824374338364716470310, 0.54128679824374338364716470310, 1.71875524449671354734302091813, 2.09989410743420151210019375166, 3.16123559013333408038601033295, 3.84827417093282664065670377297, 4.37683703969393919930731561558, 5.00570808115154024738561017517, 5.80721919713589773528495381596, 5.84985551904905678596625356553, 6.76704944160135269780785338037, 7.39109806090272817479364518718, 7.72570470739018124956570362382, 8.198969339855747519934614788121, 8.799322300174788621536040335678, 9.132515518959106249642898717489, 9.882038112594304649892676144748, 10.11595206648191875139813179680, 10.83158076103272159622626065689, 11.08809667607036098817194099997, 11.50905026458588040193845583089

Graph of the ZZ-function along the critical line