Properties

Label 16-300e8-1.1-c1e8-0-8
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

\(L(1)\) \(\approx\) \(5.432290639\)
\(L(\frac12)\) \(\approx\) \(5.432290639\)
\(L(\frac{3}{2})\) 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

−5.39966334786721538441851554985, −4.97637969915031303869463374087, −4.93455500706823304270614398985, −4.71745284857525853345750857197, −4.70433408026730913379935274600, −4.66002522645508902310423901210, −4.63842603732416048319774398786, −4.15779414298777680929023556903, −4.06911851531989675583786129916, −3.80052576868758004327729803635, −3.41108022194449428665309711009, −3.29619556753170535485728675153, −3.22439725721287402853970558972, −3.17416933566378841502331827245, −3.15591942293845268274111660768, −3.12776389750713360144645410640, −2.58457900479375986576438811531, −2.29727584641185194499338909497, −2.25269936065356816363764482463, −1.83813186662703678092373766035, −1.78048616440873034354294552684, −1.77039408276890328389779016259, −1.30192589175606002282227612411, −0.953076200547883476621943553750, −0.56835596978907116129257804668, 0.56835596978907116129257804668, 0.953076200547883476621943553750, 1.30192589175606002282227612411, 1.77039408276890328389779016259, 1.78048616440873034354294552684, 1.83813186662703678092373766035, 2.25269936065356816363764482463, 2.29727584641185194499338909497, 2.58457900479375986576438811531, 3.12776389750713360144645410640, 3.15591942293845268274111660768, 3.17416933566378841502331827245, 3.22439725721287402853970558972, 3.29619556753170535485728675153, 3.41108022194449428665309711009, 3.80052576868758004327729803635, 4.06911851531989675583786129916, 4.15779414298777680929023556903, 4.63842603732416048319774398786, 4.66002522645508902310423901210, 4.70433408026730913379935274600, 4.71745284857525853345750857197, 4.93455500706823304270614398985, 4.97637969915031303869463374087, 5.39966334786721538441851554985

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

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