Properties

Label 16-300e8-1.1-c1e8-0-0
Degree $16$
Conductor $6.561\times 10^{19}$
Sign $1$
Analytic cond. $1084.39$
Root an. cond. $1.54774$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 4·16-s − 16·19-s − 16·41-s + 64·59-s + 24·61-s + 16·64-s + 64·76-s − 32·79-s − 2·81-s + 16·101-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 64·164-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·4-s + 16-s − 3.67·19-s − 2.49·41-s + 8.33·59-s + 3.07·61-s + 2·64-s + 7.34·76-s − 3.60·79-s − 2/9·81-s + 1.59·101-s + 6.54·121-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 + 0.0798·157-s + 0.0783·163-s + 4.99·164-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1084.39\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.05940581789\)
\(L(\frac12)\) \(\approx\) \(0.05940581789\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
3 \( ( 1 + T^{4} )^{2} \)
5 \( 1 \)
good7 \( 1 - 2 p^{2} T^{4} + 7011 T^{8} - 2 p^{6} T^{12} + p^{8} T^{16} \)
11 \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \)
13 \( 1 - 194 T^{4} + 171 p^{2} T^{8} - 194 p^{4} T^{12} + p^{8} T^{16} \)
17 \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
19 \( ( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
23 \( 1 + 68 T^{4} - 434490 T^{8} + 68 p^{4} T^{12} + p^{8} T^{16} \)
29 \( ( 1 - 60 T^{2} + 1814 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 110 T^{2} + 4899 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( 1 + 868 T^{4} + 2707878 T^{8} + 868 p^{4} T^{12} + p^{8} T^{16} \)
41 \( ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
43 \( 1 + 3358 T^{4} + 8633475 T^{8} + 3358 p^{4} T^{12} + p^{8} T^{16} \)
47 \( 1 + 5188 T^{4} + 13723398 T^{8} + 5188 p^{4} T^{12} + p^{8} T^{16} \)
53 \( 1 + 5348 T^{4} + 16710438 T^{8} + 5348 p^{4} T^{12} + p^{8} T^{16} \)
59 \( ( 1 - 16 T + 170 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \)
61 \( ( 1 - 3 T + p T^{2} )^{8} \)
67 \( 1 - 6818 T^{4} + 25780611 T^{8} - 6818 p^{4} T^{12} + p^{8} T^{16} \)
71 \( ( 1 - 188 T^{2} + 17190 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 6242 T^{4} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
83 \( 1 - 284 T^{4} - 90968346 T^{8} - 284 p^{4} T^{12} + p^{8} T^{16} \)
89 \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{4} \)
97 \( 1 - 3842 T^{4} + 131200515 T^{8} - 3842 p^{4} T^{12} + p^{8} T^{16} \)
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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

−5.23688680675532335963240735048, −5.03934894693973852684617323264, −4.96095121256671559131796653808, −4.93870917422369993105520452417, −4.70488089812656058169457790238, −4.47427741858996139811691191966, −4.34424633014078671231037675409, −4.10618377130692372675998514176, −4.01907419273732841686654262399, −3.84293031434998627054576700185, −3.82709420516811561868415359340, −3.75480238412235295769257228767, −3.67359156283104065706766136618, −3.20509406619514771883719907166, −3.00907501542610901833139583488, −2.88190481429644464415862948045, −2.33696543314545808332484879466, −2.31723318336624174704414250190, −2.25556464027707890346442398374, −2.19322927879178443119481888956, −1.92753299745180275473370532683, −1.40934423812448630895860693736, −1.01698656125649746813744628889, −0.803625331949987604251320135723, −0.085887207855476897652543601301, 0.085887207855476897652543601301, 0.803625331949987604251320135723, 1.01698656125649746813744628889, 1.40934423812448630895860693736, 1.92753299745180275473370532683, 2.19322927879178443119481888956, 2.25556464027707890346442398374, 2.31723318336624174704414250190, 2.33696543314545808332484879466, 2.88190481429644464415862948045, 3.00907501542610901833139583488, 3.20509406619514771883719907166, 3.67359156283104065706766136618, 3.75480238412235295769257228767, 3.82709420516811561868415359340, 3.84293031434998627054576700185, 4.01907419273732841686654262399, 4.10618377130692372675998514176, 4.34424633014078671231037675409, 4.47427741858996139811691191966, 4.70488089812656058169457790238, 4.93870917422369993105520452417, 4.96095121256671559131796653808, 5.03934894693973852684617323264, 5.23688680675532335963240735048

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

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