↑

Properties

Label 16-75e16-1.1-c1e8-0-12
Degree $16$
Conductor $1.002\times 10^{30}$
Sign $1$
Analytic cond. $1.65652\times 10^{13}$
Root an. cond. $6.70192$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $8$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5·4-s − 16·11-s + 9·16-s + 10·19-s − 20·29-s + 16·31-s − 26·41-s + 80·44-s − 35·49-s − 30·59-s + 6·61-s − 10·64-s − 46·71-s − 50·76-s + 10·79-s − 30·89-s − 26·101-s − 10·109-s + 100·116-s + 56·121-s − 80·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 5/2·4-s − 4.82·11-s + 9/4·16-s + 2.29·19-s − 3.71·29-s + 2.87·31-s − 4.06·41-s + 12.0·44-s − 5·49-s − 3.90·59-s + 0.768·61-s − 5/4·64-s − 5.45·71-s − 5.73·76-s + 1.12·79-s − 3.17·89-s − 2.58·101-s − 0.957·109-s + 9.28·116-s + 5.09·121-s − 7.18·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(1.65652\times 10^{13}\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(8\)
Selberg data: \((16,\ 3^{16} \cdot 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + 5 T^{2} + p^{4} T^{4} + 45 T^{6} + 101 T^{8} + 45 p^{2} T^{10} + p^{8} T^{12} + 5 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 + 5 p T^{2} + 611 T^{4} + 7045 T^{6} + 57976 T^{8} + 7045 p^{2} T^{10} + 611 p^{4} T^{12} + 5 p^{7} T^{14} + p^{8} T^{16} \)
11 \( ( 1 + 2 T + p T^{2} )^{8} \)
13 \( 1 + 90 T^{2} + 3671 T^{4} + 89140 T^{6} + 1415181 T^{8} + 89140 p^{2} T^{10} + 3671 p^{4} T^{12} + 90 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 + 95 T^{2} + 4351 T^{4} + 125745 T^{6} + 2528816 T^{8} + 125745 p^{2} T^{10} + 4351 p^{4} T^{12} + 95 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - 5 T + 71 T^{2} - 255 T^{3} + 1956 T^{4} - 255 p T^{5} + 71 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 + 150 T^{2} + 10421 T^{4} + 438710 T^{6} + 12245916 T^{8} + 438710 p^{2} T^{10} + 10421 p^{4} T^{12} + 150 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 10 T + 111 T^{2} + 20 p T^{3} + 4061 T^{4} + 20 p^{2} T^{5} + 111 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 8 T + 83 T^{2} - 416 T^{3} + 3180 T^{4} - 416 p T^{5} + 83 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 + 5 p T^{2} + 17246 T^{4} + 28220 p T^{6} + 45156381 T^{8} + 28220 p^{3} T^{10} + 17246 p^{4} T^{12} + 5 p^{7} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 13 T + 183 T^{2} + 1451 T^{3} + 11760 T^{4} + 1451 p T^{5} + 183 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 + 5 p T^{2} + 22911 T^{4} + 1578205 T^{6} + 78597176 T^{8} + 1578205 p^{2} T^{10} + 22911 p^{4} T^{12} + 5 p^{7} T^{14} + p^{8} T^{16} \)
47 \( 1 + 5 p T^{2} + 26751 T^{4} + 1970705 T^{6} + 106163336 T^{8} + 1970705 p^{2} T^{10} + 26751 p^{4} T^{12} + 5 p^{7} T^{14} + p^{8} T^{16} \)
53 \( 1 + 185 T^{2} + 23006 T^{4} + 1859700 T^{6} + 115939701 T^{8} + 1859700 p^{2} T^{10} + 23006 p^{4} T^{12} + 185 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 15 T + 241 T^{2} + 2025 T^{3} + 19456 T^{4} + 2025 p T^{5} + 241 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 3 T + 98 T^{2} - 786 T^{3} + 4855 T^{4} - 786 p T^{5} + 98 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 360 T^{2} + 62716 T^{4} + 6991000 T^{6} + 550222886 T^{8} + 6991000 p^{2} T^{10} + 62716 p^{4} T^{12} + 360 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 23 T + 383 T^{2} + 4101 T^{3} + 39380 T^{4} + 4101 p T^{5} + 383 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 + 505 T^{2} + 114686 T^{4} + 15492260 T^{6} + 1375657261 T^{8} + 15492260 p^{2} T^{10} + 114686 p^{4} T^{12} + 505 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 5 T + 121 T^{2} - 15 p T^{3} + 12416 T^{4} - 15 p^{2} T^{5} + 121 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 290 T^{2} + 41701 T^{4} + 4874870 T^{6} + 471491716 T^{8} + 4874870 p^{2} T^{10} + 41701 p^{4} T^{12} + 290 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 15 T + 321 T^{2} + 3475 T^{3} + 42476 T^{4} + 3475 p T^{5} + 321 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 + 110 T^{2} + 22811 T^{4} + 2099540 T^{6} + 303117741 T^{8} + 2099540 p^{2} T^{10} + 22811 p^{4} T^{12} + 110 p^{6} T^{14} + p^{8} T^{16} \)
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

−3.96356510255872372652515346595, −3.41698299026294206162138482734, −3.34914745794590019633147368615, −3.34187432488056250526484234457, −3.28505515300618401974192921803, −3.26525649545507544409615378497, −3.19674318074098773502950715540, −3.01569087737105795653655974068, −2.78646945339686800204560620090, −2.78403862908677812666811036853, −2.75871498557569397460265781379, −2.67310573051708308121877083199, −2.62802192399007126155552800489, −2.21933883559170600081099530277, −2.14666020963079229262910118541, −2.04855885656078612067051512535, −2.02366246284975745528017868592, −1.77155652799485472279923504084, −1.77140114968897099580146515918, −1.34471140704376464070132175967, −1.27804898110608261963080019810, −1.21132093309657697175494196639, −1.13450562544769550004954571627, −1.10552392187807172596628266991, −1.07364667626898514625881126734, 0, 0, 0, 0, 0, 0, 0, 0, 1.07364667626898514625881126734, 1.10552392187807172596628266991, 1.13450562544769550004954571627, 1.21132093309657697175494196639, 1.27804898110608261963080019810, 1.34471140704376464070132175967, 1.77140114968897099580146515918, 1.77155652799485472279923504084, 2.02366246284975745528017868592, 2.04855885656078612067051512535, 2.14666020963079229262910118541, 2.21933883559170600081099530277, 2.62802192399007126155552800489, 2.67310573051708308121877083199, 2.75871498557569397460265781379, 2.78403862908677812666811036853, 2.78646945339686800204560620090, 3.01569087737105795653655974068, 3.19674318074098773502950715540, 3.26525649545507544409615378497, 3.28505515300618401974192921803, 3.34187432488056250526484234457, 3.34914745794590019633147368615, 3.41698299026294206162138482734, 3.96356510255872372652515346595

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

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