Properties

Label 4-1280e2-1.1-c3e2-0-8
Degree 44
Conductor 16384001638400
Sign 11
Analytic cond. 5703.635703.63
Root an. cond. 8.690368.69036
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  + 8·7-s − 10·9-s + 132·17-s − 264·23-s − 25·25-s + 304·31-s + 876·41-s − 408·47-s − 638·49-s − 80·63-s − 864·71-s − 724·73-s − 320·79-s − 629·81-s − 1.62e3·89-s + 2.21e3·97-s − 2.07e3·113-s + 1.05e3·119-s + 2.51e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.32e3·153-s + 157-s + ⋯
L(s)  = 1  + 0.431·7-s − 0.370·9-s + 1.88·17-s − 2.39·23-s − 1/5·25-s + 1.76·31-s + 3.33·41-s − 1.26·47-s − 1.86·49-s − 0.159·63-s − 1.44·71-s − 1.16·73-s − 0.455·79-s − 0.862·81-s − 1.92·89-s + 2.31·97-s − 1.72·113-s + 0.813·119-s + 1.89·121-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 − 0.697·153-s + 0.000508·157-s + ⋯

Functional equation

Λ(s)=(1638400s/2ΓC(s)2L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1638400s/2ΓC(s+3/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 16384001638400    =    216522^{16} \cdot 5^{2}
Sign: 11
Analytic conductor: 5703.635703.63
Root analytic conductor: 8.690368.69036
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1638400, ( :3/2,3/2), 1)(4,\ 1638400,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) \approx 2.4334664732.433466473
L(12)L(\frac12) \approx 2.4334664732.433466473
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
5C2C_2 1+p2T2 1 + p^{2} T^{2}
good3C22C_2^2 1+10T2+p6T4 1 + 10 T^{2} + p^{6} T^{4}
7C2C_2 (14T+p3T2)2 ( 1 - 4 T + p^{3} T^{2} )^{2}
11C22C_2^2 12518T2+p6T4 1 - 2518 T^{2} + p^{6} T^{4}
13C22C_2^2 11030T2+p6T4 1 - 1030 T^{2} + p^{6} T^{4}
17C2C_2 (166T+p3T2)2 ( 1 - 66 T + p^{3} T^{2} )^{2}
19C22C_2^2 13718T2+p6T4 1 - 3718 T^{2} + p^{6} T^{4}
23C2C_2 (1+132T+p3T2)2 ( 1 + 132 T + p^{3} T^{2} )^{2}
29C22C_2^2 140678T2+p6T4 1 - 40678 T^{2} + p^{6} T^{4}
31C2C_2 (1152T+p3T2)2 ( 1 - 152 T + p^{3} T^{2} )^{2}
37C22C_2^2 1100150T2+p6T4 1 - 100150 T^{2} + p^{6} T^{4}
41C2C_2 (1438T+p3T2)2 ( 1 - 438 T + p^{3} T^{2} )^{2}
43C22C_2^2 1157990T2+p6T4 1 - 157990 T^{2} + p^{6} T^{4}
47C2C_2 (1+204T+p3T2)2 ( 1 + 204 T + p^{3} T^{2} )^{2}
53C22C_2^2 1248470T2+p6T4 1 - 248470 T^{2} + p^{6} T^{4}
59C22C_2^2 1234358T2+p6T4 1 - 234358 T^{2} + p^{6} T^{4}
61C22C_2^2 1+359642T2+p6T4 1 + 359642 T^{2} + p^{6} T^{4}
67C22C_2^2 1+447050T2+p6T4 1 + 447050 T^{2} + p^{6} T^{4}
71C2C_2 (1+432T+p3T2)2 ( 1 + 432 T + p^{3} T^{2} )^{2}
73C2C_2 (1+362T+p3T2)2 ( 1 + 362 T + p^{3} T^{2} )^{2}
79C2C_2 (1+160T+p3T2)2 ( 1 + 160 T + p^{3} T^{2} )^{2}
83C22C_2^2 11138390T2+p6T4 1 - 1138390 T^{2} + p^{6} T^{4}
89C2C_2 (1+810T+p3T2)2 ( 1 + 810 T + p^{3} T^{2} )^{2}
97C2C_2 (11106T+p3T2)2 ( 1 - 1106 T + p^{3} T^{2} )^{2}
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   L(s)=p j=14(1αj,pps)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

−9.922209730572301062251873999987, −9.258253447272109001920048394046, −8.390003232217146344175375369858, −8.310445717514221102856939796935, −7.964986002609789878754630899586, −7.46739943291890699445640228728, −7.32260100981730608015028871905, −6.30180583583198474534969304251, −6.25028994028918074745659101015, −5.70296392808869601463585874376, −5.53260172383680491013297982608, −4.63203746638670448118391121580, −4.46099514712251998840508123575, −3.96525022484581501747524334019, −3.20304566533653100763421543424, −2.98284293069613874450983070645, −2.25801337125959235671303844648, −1.65595392011314236820540630010, −1.09443746676811413137401110765, −0.39504804746985530018064498754, 0.39504804746985530018064498754, 1.09443746676811413137401110765, 1.65595392011314236820540630010, 2.25801337125959235671303844648, 2.98284293069613874450983070645, 3.20304566533653100763421543424, 3.96525022484581501747524334019, 4.46099514712251998840508123575, 4.63203746638670448118391121580, 5.53260172383680491013297982608, 5.70296392808869601463585874376, 6.25028994028918074745659101015, 6.30180583583198474534969304251, 7.32260100981730608015028871905, 7.46739943291890699445640228728, 7.964986002609789878754630899586, 8.310445717514221102856939796935, 8.390003232217146344175375369858, 9.258253447272109001920048394046, 9.922209730572301062251873999987

Graph of the ZZ-function along the critical line