Properties

Label 16-570e8-1.1-c1e8-0-4
Degree $16$
Conductor $1.114\times 10^{22}$
Sign $1$
Analytic cond. $184168.$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 2·3-s + 36·4-s − 16·6-s + 4·7-s − 120·8-s + 3·9-s + 72·12-s − 32·14-s + 330·16-s − 24·18-s + 12·19-s + 8·21-s − 240·24-s − 4·25-s − 2·27-s + 144·28-s + 4·29-s − 792·32-s + 108·36-s − 96·38-s − 16·41-s − 64·42-s − 40·43-s + 660·48-s − 18·49-s + 32·50-s + ⋯
L(s)  = 1  − 5.65·2-s + 1.15·3-s + 18·4-s − 6.53·6-s + 1.51·7-s − 42.4·8-s + 9-s + 20.7·12-s − 8.55·14-s + 82.5·16-s − 5.65·18-s + 2.75·19-s + 1.74·21-s − 48.9·24-s − 4/5·25-s − 0.384·27-s + 27.2·28-s + 0.742·29-s − 140.·32-s + 18·36-s − 15.5·38-s − 2.49·41-s − 9.87·42-s − 6.09·43-s + 95.2·48-s − 2.57·49-s + 4.52·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(184168.\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2533015051\)
\(L(\frac12)\) \(\approx\) \(0.2533015051\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{8} \)
3 \( 1 - 2 T + T^{2} + 2 p T^{3} - 20 T^{4} + 2 p^{2} T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
5 \( ( 1 + T^{2} )^{4} \)
19 \( 1 - 12 T + 80 T^{2} - 396 T^{3} + 1678 T^{4} - 396 p T^{5} + 80 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
good7 \( ( 1 - 2 T + 15 T^{2} - 46 T^{3} + 114 T^{4} - 46 p T^{5} + 15 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \)
13 \( 1 - 66 T^{2} + 2033 T^{4} - 40002 T^{6} + 586036 T^{8} - 40002 p^{2} T^{10} + 2033 p^{4} T^{12} - 66 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 70 T^{2} + 2429 T^{4} - 56594 T^{6} + 1041112 T^{8} - 56594 p^{2} T^{10} + 2429 p^{4} T^{12} - 70 p^{6} T^{14} + p^{8} T^{16} \)
23 \( 1 - 90 T^{2} + 4601 T^{4} - 164426 T^{6} + 4341492 T^{8} - 164426 p^{2} T^{10} + 4601 p^{4} T^{12} - 90 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - 2 T + 105 T^{2} - 150 T^{3} + 4400 T^{4} - 150 p T^{5} + 105 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 - 148 T^{2} + 11560 T^{4} - 587580 T^{6} + 21376334 T^{8} - 587580 p^{2} T^{10} + 11560 p^{4} T^{12} - 148 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 - 88 T^{2} + 3228 T^{4} - 29864 T^{6} - 1445338 T^{8} - 29864 p^{2} T^{10} + 3228 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 8 T + 162 T^{2} + 960 T^{3} + 9938 T^{4} + 960 p T^{5} + 162 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 + 20 T + 260 T^{2} + 2468 T^{3} + 18486 T^{4} + 2468 p T^{5} + 260 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - 248 T^{2} + 31100 T^{4} - 2491464 T^{6} + 138964998 T^{8} - 2491464 p^{2} T^{10} + 31100 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 + 14 T + 245 T^{2} + 2058 T^{3} + 376 p T^{4} + 2058 p T^{5} + 245 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 2 T + 75 T^{2} - 74 T^{3} + 34 p T^{4} - 74 p T^{5} + 75 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 8 T + 104 T^{2} - 616 T^{3} + 8350 T^{4} - 616 p T^{5} + 104 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 26 T^{2} + 3473 T^{4} - 490082 T^{6} + 11677428 T^{8} - 490082 p^{2} T^{10} + 3473 p^{4} T^{12} - 26 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 12 T + 188 T^{2} + 860 T^{3} + 11174 T^{4} + 860 p T^{5} + 188 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 2 T + 173 T^{2} - 326 T^{3} + 13372 T^{4} - 326 p T^{5} + 173 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 - 412 T^{2} + 86600 T^{4} - 11652692 T^{6} + 1092874510 T^{8} - 11652692 p^{2} T^{10} + 86600 p^{4} T^{12} - 412 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 - 124 T^{2} + 17048 T^{4} - 1000820 T^{6} + 99135310 T^{8} - 1000820 p^{2} T^{10} + 17048 p^{4} T^{12} - 124 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 44 T + 1054 T^{2} - 16468 T^{3} + 183154 T^{4} - 16468 p T^{5} + 1054 p^{2} T^{6} - 44 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 656 T^{2} + 197852 T^{4} - 35779696 T^{6} + 4238516038 T^{8} - 35779696 p^{2} T^{10} + 197852 p^{4} T^{12} - 656 p^{6} T^{14} + p^{8} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.88046438051183659199395696710, −4.69727786318148801179807807166, −4.49409032614249574578578552824, −4.33417152981229313593279279001, −3.84585071863623973304417637436, −3.73156811991292069623518671254, −3.56229846452421221385664552167, −3.40745057945928375366645701001, −3.34354675687054925249476296885, −3.16892241757326770472515507586, −3.15576146990859265651399961220, −3.08358704447890703485588413940, −2.92464577046317464303280717704, −2.56196079785262663743072419009, −2.27382941578790942647362525781, −2.01072518769825544870660576035, −1.84891280229931707191790880047, −1.77863121993985399373691587974, −1.77745742364231431421247865690, −1.69806550904465297243744247141, −1.58627946287709440635346184145, −1.12889941610199753202969980449, −0.856071556048657012534188796177, −0.62111438818050233261484711067, −0.24554814187941296673115981506, 0.24554814187941296673115981506, 0.62111438818050233261484711067, 0.856071556048657012534188796177, 1.12889941610199753202969980449, 1.58627946287709440635346184145, 1.69806550904465297243744247141, 1.77745742364231431421247865690, 1.77863121993985399373691587974, 1.84891280229931707191790880047, 2.01072518769825544870660576035, 2.27382941578790942647362525781, 2.56196079785262663743072419009, 2.92464577046317464303280717704, 3.08358704447890703485588413940, 3.15576146990859265651399961220, 3.16892241757326770472515507586, 3.34354675687054925249476296885, 3.40745057945928375366645701001, 3.56229846452421221385664552167, 3.73156811991292069623518671254, 3.84585071863623973304417637436, 4.33417152981229313593279279001, 4.49409032614249574578578552824, 4.69727786318148801179807807166, 4.88046438051183659199395696710

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.