Properties

Label 4-294e2-1.1-c5e2-0-25
Degree 44
Conductor 8643686436
Sign 11
Analytic cond. 2223.392223.39
Root an. cond. 6.866796.86679
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 18·3-s + 48·4-s − 18·5-s − 144·6-s + 256·8-s + 243·9-s − 144·10-s + 2·11-s − 864·12-s − 288·13-s + 324·15-s + 1.28e3·16-s − 1.53e3·17-s + 1.94e3·18-s − 1.18e3·19-s − 864·20-s + 16·22-s + 3.39e3·23-s − 4.60e3·24-s − 1.30e3·25-s − 2.30e3·26-s − 2.91e3·27-s − 3.97e3·29-s + 2.59e3·30-s − 7.59e3·31-s + 6.14e3·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.321·5-s − 1.63·6-s + 1.41·8-s + 9-s − 0.455·10-s + 0.00498·11-s − 1.73·12-s − 0.472·13-s + 0.371·15-s + 5/4·16-s − 1.28·17-s + 1.41·18-s − 0.754·19-s − 0.482·20-s + 0.00704·22-s + 1.33·23-s − 1.63·24-s − 0.416·25-s − 0.668·26-s − 0.769·27-s − 0.877·29-s + 0.525·30-s − 1.41·31-s + 1.06·32-s + ⋯

Functional equation

Λ(s)=(86436s/2ΓC(s)2L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(86436s/2ΓC(s+5/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8643686436    =    2232742^{2} \cdot 3^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 2223.392223.39
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 86436, ( :5/2,5/2), 1)(4,\ 86436,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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)
bad2C1C_1 (1p2T)2 ( 1 - p^{2} T )^{2}
3C1C_1 (1+p2T)2 ( 1 + p^{2} T )^{2}
7 1 1
good5D4D_{4} 1+18T+1626T2+18p5T3+p10T4 1 + 18 T + 1626 T^{2} + 18 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 12T59002T22p5T3+p10T4 1 - 2 T - 59002 T^{2} - 2 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1+288T+688042T2+288p5T3+p10T4 1 + 288 T + 688042 T^{2} + 288 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 1+90pT+2366314T2+90p6T3+p10T4 1 + 90 p T + 2366314 T^{2} + 90 p^{6} T^{3} + p^{10} T^{4}
19D4D_{4} 1+1188T+4834534T2+1188p5T3+p10T4 1 + 1188 T + 4834534 T^{2} + 1188 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 13390T+6218086T23390p5T3+p10T4 1 - 3390 T + 6218086 T^{2} - 3390 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1+3976T+31254662T2+3976p5T3+p10T4 1 + 3976 T + 31254662 T^{2} + 3976 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 1+7596T+61727326T2+7596p5T3+p10T4 1 + 7596 T + 61727326 T^{2} + 7596 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 12688T+126774470T22688p5T3+p10T4 1 - 2688 T + 126774470 T^{2} - 2688 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 1+36630T+559995322T2+36630p5T3+p10T4 1 + 36630 T + 559995322 T^{2} + 36630 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1+23032T+329072262T2+23032p5T3+p10T4 1 + 23032 T + 329072262 T^{2} + 23032 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1+864T+46643358T2+864p5T3+p10T4 1 + 864 T + 46643358 T^{2} + 864 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 1+32920T+983844566T2+32920p5T3+p10T4 1 + 32920 T + 983844566 T^{2} + 32920 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 126712T+697343334T226712p5T3+p10T4 1 - 26712 T + 697343334 T^{2} - 26712 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 1+20412T+1230091258T2+20412p5T3+p10T4 1 + 20412 T + 1230091258 T^{2} + 20412 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 1+36172T+33148290pT2+36172p5T3+p10T4 1 + 36172 T + 33148290 p T^{2} + 36172 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 173706T+3356433686T273706p5T3+p10T4 1 - 73706 T + 3356433686 T^{2} - 73706 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 174772T+3993671602T274772p5T3+p10T4 1 - 74772 T + 3993671602 T^{2} - 74772 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 1+23116T+6249589662T2+23116p5T3+p10T4 1 + 23116 T + 6249589662 T^{2} + 23116 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 1+147816T+13316083030T2+147816p5T3+p10T4 1 + 147816 T + 13316083030 T^{2} + 147816 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1+164646T+17878567722T2+164646p5T3+p10T4 1 + 164646 T + 17878567722 T^{2} + 164646 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1+162036T+20528488258T2+162036p5T3+p10T4 1 + 162036 T + 20528488258 T^{2} + 162036 p^{5} T^{3} + p^{10} T^{4}
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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

−10.94051470496383352451219154701, −10.62306998003108148428554252806, −9.810306264878780583963266579585, −9.603927862172053627915586273507, −8.553666883184945449629858081451, −8.362188324449009716368627233228, −7.29526775415216150857973534705, −7.14889230383366722223268502247, −6.50884808899763086003673909495, −6.35243652157073466088126387616, −5.31144498956551657011629354982, −5.24575952664842368545525495579, −4.71043719111555090532257570228, −4.06735968326835452851113059023, −3.56773574337593234013412674276, −2.85417082178007813404195012162, −1.88309949483789137823819999482, −1.52483767391626227590160960001, 0, 0, 1.52483767391626227590160960001, 1.88309949483789137823819999482, 2.85417082178007813404195012162, 3.56773574337593234013412674276, 4.06735968326835452851113059023, 4.71043719111555090532257570228, 5.24575952664842368545525495579, 5.31144498956551657011629354982, 6.35243652157073466088126387616, 6.50884808899763086003673909495, 7.14889230383366722223268502247, 7.29526775415216150857973534705, 8.362188324449009716368627233228, 8.553666883184945449629858081451, 9.603927862172053627915586273507, 9.810306264878780583963266579585, 10.62306998003108148428554252806, 10.94051470496383352451219154701

Graph of the ZZ-function along the critical line