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Properties

Label 8-48e4-1.1-c16e4-0-1
Degree 88
Conductor 53084165308416
Sign 11
Analytic cond. 3.68554×1073.68554\times 10^{7}
Root an. cond. 8.826998.82699
Motivic weight 1616
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.40e5·5-s − 2.86e7·9-s − 5.86e8·13-s − 5.71e8·17-s − 3.86e11·25-s − 5.85e11·29-s − 2.77e12·37-s − 9.74e12·41-s + 4.02e12·45-s + 8.88e13·49-s + 9.39e13·53-s + 3.74e14·61-s + 8.23e13·65-s + 1.37e15·73-s + 6.17e14·81-s + 8.01e13·85-s + 2.06e16·89-s + 4.57e16·97-s + 3.93e16·101-s + 5.91e16·109-s + 1.24e17·113-s + 1.68e16·117-s + 1.63e17·121-s + 6.16e16·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.359·5-s − 2/3·9-s − 0.719·13-s − 0.0818·17-s − 2.53·25-s − 1.17·29-s − 0.788·37-s − 1.22·41-s + 0.239·45-s + 2.67·49-s + 1.50·53-s + 1.95·61-s + 0.258·65-s + 1.71·73-s + 1/3·81-s + 0.0294·85-s + 5.24·89-s + 5.83·97-s + 3.62·101-s + 2.96·109-s + 4.67·113-s + 0.479·117-s + 3.56·121-s + 1.03·125-s + 0.420·145-s + ⋯

Functional equation

Λ(s)=(5308416s/2 ΓC(s)4 L(s)=( Λ(17−s)\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}
Λ(s)=(5308416s/2 ΓC(s+8)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+8)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 88
Conductor: 53084165308416    =    216⋅342^{16} \cdot 3^{4}
Sign: 11
Analytic conductor: 3.68554×1073.68554\times 10^{7}
Root analytic conductor: 8.826998.82699
Motivic weight: 1616
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 5308416, ( :8,8,8,8), 1)(8,\ 5308416,\ (\ :8, 8, 8, 8),\ 1)

Particular Values

L(172)L(\frac{17}{2}) ≈\approx 2.9888391232.988839123
L(12)L(\frac12) ≈\approx 2.9888391232.988839123
L(9)L(9) 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+p15T2)2 ( 1 + p^{15} T^{2} )^{2}
good5D4D_{4} (1+70212T+40121922958pT2+70212p16T3+p32T4)2 ( 1 + 70212 T + 40121922958 p T^{2} + 70212 p^{16} T^{3} + p^{32} T^{4} )^{2}
7D4×C2D_4\times C_2 1−1813451510116p2T2+ 1 - 1813451510116 p^{2} T^{2} + 35 ⁣⋯ ⁣5835\!\cdots\!58p6T4−1813451510116p34T6+p64T8 p^{6} T^{4} - 1813451510116 p^{34} T^{6} + p^{64} T^{8}
11D4×C2D_4\times C_2 1−163743480736580260T2+ 1 - 163743480736580260 T^{2} + 90 ⁣⋯ ⁣0290\!\cdots\!02p2T4−163743480736580260p32T6+p64T8 p^{2} T^{4} - 163743480736580260 p^{32} T^{6} + p^{64} T^{8}
13D4D_{4} (1+1735756p2T+6679213418435574p2T2+1735756p18T3+p32T4)2 ( 1 + 1735756 p^{2} T + 6679213418435574 p^{2} T^{2} + 1735756 p^{18} T^{3} + p^{32} T^{4} )^{2}
17D4D_{4} (1+285502668T+55375760720126595494T2+285502668p16T3+p32T4)2 ( 1 + 285502668 T + 55375760720126595494 T^{2} + 285502668 p^{16} T^{3} + p^{32} T^{4} )^{2}
19D4×C2D_4\times C_2 1− 1 - 10 ⁣⋯ ⁣6410\!\cdots\!64T2+ T^{2} + 44 ⁣⋯ ⁣6244\!\cdots\!62T4− T^{4} - 10 ⁣⋯ ⁣6410\!\cdots\!64p32T6+p64T8 p^{32} T^{6} + p^{64} T^{8}
23D4×C2D_4\times C_2 1− 1 - 14 ⁣⋯ ⁣5214\!\cdots\!52T2+ T^{2} + 12 ⁣⋯ ⁣0212\!\cdots\!02T4− T^{4} - 14 ⁣⋯ ⁣5214\!\cdots\!52p32T6+p64T8 p^{32} T^{6} + p^{64} T^{8}
29D4D_{4} (1+292810708212T+ ( 1 + 292810708212 T + 51 ⁣⋯ ⁣2251\!\cdots\!22T2+292810708212p16T3+p32T4)2 T^{2} + 292810708212 p^{16} T^{3} + p^{32} T^{4} )^{2}
31D4×C2D_4\times C_2 1− 1 - 17 ⁣⋯ ⁣8017\!\cdots\!80T2+ T^{2} + 14 ⁣⋯ ⁣2214\!\cdots\!22T4− T^{4} - 17 ⁣⋯ ⁣8017\!\cdots\!80p32T6+p64T8 p^{32} T^{6} + p^{64} T^{8}
37D4D_{4} (1+1385235255548T+ ( 1 + 1385235255548 T + 14 ⁣⋯ ⁣2214\!\cdots\!22T2+1385235255548p16T3+p32T4)2 T^{2} + 1385235255548 p^{16} T^{3} + p^{32} T^{4} )^{2}
41D4D_{4} (1+4872348556812T+ ( 1 + 4872348556812 T + 62 ⁣⋯ ⁣9462\!\cdots\!94T2+4872348556812p16T3+p32T4)2 T^{2} + 4872348556812 p^{16} T^{3} + p^{32} T^{4} )^{2}
43D4×C2D_4\times C_2 1− 1 - 31 ⁣⋯ ⁣0431\!\cdots\!04T2+ T^{2} + 61 ⁣⋯ ⁣2261\!\cdots\!22T4− T^{4} - 31 ⁣⋯ ⁣0431\!\cdots\!04p32T6+p64T8 p^{32} T^{6} + p^{64} T^{8}
47D4×C2D_4\times C_2 1− 1 - 21 ⁣⋯ ⁣6421\!\cdots\!64T2+ T^{2} + 17 ⁣⋯ ⁣0217\!\cdots\!02T4− T^{4} - 21 ⁣⋯ ⁣6421\!\cdots\!64p32T6+p64T8 p^{32} T^{6} + p^{64} T^{8}
53D4D_{4} (1−46975486697100T+ ( 1 - 46975486697100 T + 77 ⁣⋯ ⁣8677\!\cdots\!86T2−46975486697100p16T3+p32T4)2 T^{2} - 46975486697100 p^{16} T^{3} + p^{32} T^{4} )^{2}
59D4×C2D_4\times C_2 1− 1 - 78 ⁣⋯ ⁣6078\!\cdots\!60T2+ T^{2} + 24 ⁣⋯ ⁣6224\!\cdots\!62T4− T^{4} - 78 ⁣⋯ ⁣6078\!\cdots\!60p32T6+p64T8 p^{32} T^{6} + p^{64} T^{8}
61D4D_{4} (1−187081173156196T+ ( 1 - 187081173156196 T + 22 ⁣⋯ ⁣0222\!\cdots\!02T2−187081173156196p16T3+p32T4)2 T^{2} - 187081173156196 p^{16} T^{3} + p^{32} T^{4} )^{2}
67D4×C2D_4\times C_2 1+ 1 + 18 ⁣⋯ ⁣9618\!\cdots\!96T2+ T^{2} + 45 ⁣⋯ ⁣2245\!\cdots\!22T4+ T^{4} + 18 ⁣⋯ ⁣9618\!\cdots\!96p32T6+p64T8 p^{32} T^{6} + p^{64} T^{8}
71D4×C2D_4\times C_2 1− 1 - 50 ⁣⋯ ⁣8450\!\cdots\!84T2+ T^{2} + 75 ⁣⋯ ⁣2275\!\cdots\!22T4− T^{4} - 50 ⁣⋯ ⁣8450\!\cdots\!84p32T6+p64T8 p^{32} T^{6} + p^{64} T^{8}
73D4D_{4} (1−689974858272260T+ ( 1 - 689974858272260 T + 10 ⁣⋯ ⁣8610\!\cdots\!86T2−689974858272260p16T3+p32T4)2 T^{2} - 689974858272260 p^{16} T^{3} + p^{32} T^{4} )^{2}
79D4×C2D_4\times C_2 1− 1 - 58 ⁣⋯ ⁣1658\!\cdots\!16T2+ T^{2} + 18 ⁣⋯ ⁣9018\!\cdots\!90T4− T^{4} - 58 ⁣⋯ ⁣1658\!\cdots\!16p32T6+p64T8 p^{32} T^{6} + p^{64} T^{8}
83D4×C2D_4\times C_2 1− 1 - 14 ⁣⋯ ⁣6814\!\cdots\!68T2+ T^{2} + 94 ⁣⋯ ⁣7894\!\cdots\!78T4− T^{4} - 14 ⁣⋯ ⁣6814\!\cdots\!68p32T6+p64T8 p^{32} T^{6} + p^{64} T^{8}
89D4D_{4} (1−10333279314316932T+ ( 1 - 10333279314316932 T + 52 ⁣⋯ ⁣7852\!\cdots\!78T2−10333279314316932p16T3+p32T4)2 T^{2} - 10333279314316932 p^{16} T^{3} + p^{32} T^{4} )^{2}
97D4D_{4} (1−22856009041316420T+ ( 1 - 22856009041316420 T + 25 ⁣⋯ ⁣5825\!\cdots\!58T2−22856009041316420p16T3+p32T4)2 T^{2} - 22856009041316420 p^{16} T^{3} + p^{32} T^{4} )^{2}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.618240043885811050711211454956, −7.64952462279412294150800208921, −7.58434566357667481498786200258, −7.45670223527929715201828235175, −7.37004042987521028905699257540, −6.74651464549200882498748123583, −6.12924831322160617795791312833, −6.02089859849640008218403654781, −5.97237937388445877514528117082, −5.45349027786787418213711074836, −4.85973798952524210455834229786, −4.77889434722127173524023625977, −4.71052432045709622590226202616, −3.73784114846193849339579340085, −3.54432838761156332739151994531, −3.53670297657709487747972958832, −3.44433223266133500126060916990, −2.23773805320163551158854854031, −2.20757922059709369257527719095, −2.12963540060450191706320775637, −2.05417949892408489313833385916, −1.07053310363622544697666395004, −0.69635231894844950237929421598, −0.61548304338691277365798794316, −0.25347216797906775683724147926, 0.25347216797906775683724147926, 0.61548304338691277365798794316, 0.69635231894844950237929421598, 1.07053310363622544697666395004, 2.05417949892408489313833385916, 2.12963540060450191706320775637, 2.20757922059709369257527719095, 2.23773805320163551158854854031, 3.44433223266133500126060916990, 3.53670297657709487747972958832, 3.54432838761156332739151994531, 3.73784114846193849339579340085, 4.71052432045709622590226202616, 4.77889434722127173524023625977, 4.85973798952524210455834229786, 5.45349027786787418213711074836, 5.97237937388445877514528117082, 6.02089859849640008218403654781, 6.12924831322160617795791312833, 6.74651464549200882498748123583, 7.37004042987521028905699257540, 7.45670223527929715201828235175, 7.58434566357667481498786200258, 7.64952462279412294150800208921, 8.618240043885811050711211454956

Graph of the ZZ-function along the critical line