Properties

Label 16-414e8-1.1-c2e8-0-2
Degree $16$
Conductor $8.630\times 10^{20}$
Sign $1$
Analytic cond. $2.62230\times 10^{8}$
Root an. cond. $3.35867$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8·4-s + 16·13-s + 40·16-s − 16·23-s + 136·25-s + 144·29-s − 128·31-s + 16·41-s + 112·47-s + 216·49-s + 128·52-s − 80·59-s + 160·64-s − 32·71-s + 64·73-s − 128·92-s + 1.08e3·100-s − 464·101-s + 1.15e3·116-s + 456·121-s − 1.02e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2·4-s + 1.23·13-s + 5/2·16-s − 0.695·23-s + 5.43·25-s + 4.96·29-s − 4.12·31-s + 0.390·41-s + 2.38·47-s + 4.40·49-s + 2.46·52-s − 1.35·59-s + 5/2·64-s − 0.450·71-s + 0.876·73-s − 1.39·92-s + 10.8·100-s − 4.59·101-s + 9.93·116-s + 3.76·121-s − 8.25·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(2.62230\times 10^{8}\)
Root analytic conductor: \(3.35867\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 23^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(43.51908124\)
\(L(\frac12)\) \(\approx\) \(43.51908124\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p T^{2} )^{4} \)
3 \( 1 \)
23 \( 1 + 16 T + 36 p T^{2} - 16 p^{2} T^{3} + 10 p^{3} T^{4} - 16 p^{4} T^{5} + 36 p^{5} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \)
good5 \( 1 - 136 T^{2} + 72 p^{3} T^{4} - 381608 T^{6} + 11301266 T^{8} - 381608 p^{4} T^{10} + 72 p^{11} T^{12} - 136 p^{12} T^{14} + p^{16} T^{16} \)
7 \( 1 - 216 T^{2} + 24168 T^{4} - 1784376 T^{6} + 99228242 T^{8} - 1784376 p^{4} T^{10} + 24168 p^{8} T^{12} - 216 p^{12} T^{14} + p^{16} T^{16} \)
11 \( 1 - 456 T^{2} + 81948 T^{4} - 625896 p T^{6} + 455741702 T^{8} - 625896 p^{5} T^{10} + 81948 p^{8} T^{12} - 456 p^{12} T^{14} + p^{16} T^{16} \)
13 \( ( 1 - 8 T + 552 T^{2} - 2920 T^{3} + 128498 T^{4} - 2920 p^{2} T^{5} + 552 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
17 \( 1 - 1144 T^{2} + 669768 T^{4} - 262214744 T^{6} + 82094771282 T^{8} - 262214744 p^{4} T^{10} + 669768 p^{8} T^{12} - 1144 p^{12} T^{14} + p^{16} T^{16} \)
19 \( 1 - 744 T^{2} + 390408 T^{4} - 167906184 T^{6} + 70000513682 T^{8} - 167906184 p^{4} T^{10} + 390408 p^{8} T^{12} - 744 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 - 72 T + 156 p T^{2} - 185976 T^{3} + 6188678 T^{4} - 185976 p^{2} T^{5} + 156 p^{5} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
31 \( ( 1 + 64 T + 4660 T^{2} + 187072 T^{3} + 7111078 T^{4} + 187072 p^{2} T^{5} + 4660 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( 1 - 3784 T^{2} + 9839196 T^{4} - 18805098488 T^{6} + 29956322122118 T^{8} - 18805098488 p^{4} T^{10} + 9839196 p^{8} T^{12} - 3784 p^{12} T^{14} + p^{16} T^{16} \)
41 \( ( 1 - 8 T + 4156 T^{2} - 30008 T^{3} + 9529414 T^{4} - 30008 p^{2} T^{5} + 4156 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
43 \( 1 - 8088 T^{2} + 34142760 T^{4} - 98670713208 T^{6} + 210334516194962 T^{8} - 98670713208 p^{4} T^{10} + 34142760 p^{8} T^{12} - 8088 p^{12} T^{14} + p^{16} T^{16} \)
47 \( ( 1 - 56 T + 8424 T^{2} - 336088 T^{3} + 27713906 T^{4} - 336088 p^{2} T^{5} + 8424 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
53 \( 1 + 1352 T^{2} + 21551880 T^{4} + 17467557352 T^{6} + 219565933670162 T^{8} + 17467557352 p^{4} T^{10} + 21551880 p^{8} T^{12} + 1352 p^{12} T^{14} + p^{16} T^{16} \)
59 \( ( 1 + 40 T + 8136 T^{2} + 353096 T^{3} + 39128402 T^{4} + 353096 p^{2} T^{5} + 8136 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
61 \( 1 - 5064 T^{2} + 6827484 T^{4} + 65858096136 T^{6} - 540796603024378 T^{8} + 65858096136 p^{4} T^{10} + 6827484 p^{8} T^{12} - 5064 p^{12} T^{14} + p^{16} T^{16} \)
67 \( 1 - 23720 T^{2} + 249032776 T^{4} - 1616057760584 T^{6} + 7951613041719442 T^{8} - 1616057760584 p^{4} T^{10} + 249032776 p^{8} T^{12} - 23720 p^{12} T^{14} + p^{16} T^{16} \)
71 \( ( 1 + 16 T + 14460 T^{2} + 389360 T^{3} + 94149446 T^{4} + 389360 p^{2} T^{5} + 14460 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
73 \( ( 1 - 32 T + 13444 T^{2} - 381536 T^{3} + 87080518 T^{4} - 381536 p^{2} T^{5} + 13444 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 - 41128 T^{2} + 784692744 T^{4} - 9037352624840 T^{6} + 68645500305242450 T^{8} - 9037352624840 p^{4} T^{10} + 784692744 p^{8} T^{12} - 41128 p^{12} T^{14} + p^{16} T^{16} \)
83 \( 1 - 40200 T^{2} + 755019036 T^{4} - 8836015681464 T^{6} + 71865614384276102 T^{8} - 8836015681464 p^{4} T^{10} + 755019036 p^{8} T^{12} - 40200 p^{12} T^{14} + p^{16} T^{16} \)
89 \( 1 - 22920 T^{2} + 396852072 T^{4} - 4291016337192 T^{6} + 40042732628956754 T^{8} - 4291016337192 p^{4} T^{10} + 396852072 p^{8} T^{12} - 22920 p^{12} T^{14} + p^{16} T^{16} \)
97 \( 1 - 45064 T^{2} + 1040347356 T^{4} - 15986808721208 T^{6} + 176417545052741318 T^{8} - 15986808721208 p^{4} T^{10} + 1040347356 p^{8} T^{12} - 45064 p^{12} T^{14} + p^{16} T^{16} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.77682654448227212549805689938, −4.55860845460856714554801963640, −4.24656236989896791309170538384, −4.24608944712933215814925298572, −4.18441402792074751195185737625, −4.00802345045925616823941882517, −3.83121069507302909208833443369, −3.63113612908461775154109433883, −3.54340088501640822184685471019, −3.06572317338829865076050093197, −3.03228089431441195018055311318, −3.00415171403197697454468938592, −2.82169245448700160981654220164, −2.72194334931495845458827669328, −2.62023018096792837983494033703, −2.51679196852960860579169542406, −1.96808299084530609912784090616, −1.94331921950573459286273299994, −1.75674905800591662987561793444, −1.60705834405515214079794036409, −1.17364445297236135583088092785, −0.974917487559090904866888459665, −0.816243225692286988036294091480, −0.76348638188769083469959916896, −0.49425205131555337700942275006, 0.49425205131555337700942275006, 0.76348638188769083469959916896, 0.816243225692286988036294091480, 0.974917487559090904866888459665, 1.17364445297236135583088092785, 1.60705834405515214079794036409, 1.75674905800591662987561793444, 1.94331921950573459286273299994, 1.96808299084530609912784090616, 2.51679196852960860579169542406, 2.62023018096792837983494033703, 2.72194334931495845458827669328, 2.82169245448700160981654220164, 3.00415171403197697454468938592, 3.03228089431441195018055311318, 3.06572317338829865076050093197, 3.54340088501640822184685471019, 3.63113612908461775154109433883, 3.83121069507302909208833443369, 4.00802345045925616823941882517, 4.18441402792074751195185737625, 4.24608944712933215814925298572, 4.24656236989896791309170538384, 4.55860845460856714554801963640, 4.77682654448227212549805689938

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

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