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Properties

Label 16-3627e8-1.1-c0e8-0-0
Degree $16$
Conductor $2.995\times 10^{28}$
Sign $1$
Analytic cond. $115.249$
Root an. cond. $1.34540$
Motivic weight $0$
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  + 4-s − 3·7-s − 13-s + 16-s − 3·19-s − 4·25-s − 3·28-s + 31-s + 2·37-s + 2·43-s + 4·49-s − 52-s + 3·61-s − 3·67-s − 2·73-s − 3·76-s − 7·79-s + 3·91-s − 5·97-s − 4·100-s + 3·103-s + 5·109-s − 3·112-s − 121-s + 124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 4-s − 3·7-s − 13-s + 16-s − 3·19-s − 4·25-s − 3·28-s + 31-s + 2·37-s + 2·43-s + 4·49-s − 52-s + 3·61-s − 3·67-s − 2·73-s − 3·76-s − 7·79-s + 3·91-s − 5·97-s − 4·100-s + 3·103-s + 5·109-s − 3·112-s − 121-s + 124-s + 127-s + 131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 13^{8} \cdot 31^{8}\)
Sign: $1$
Analytic conductor: \(115.249\)
Root analytic conductor: \(1.34540\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 13^{8} \cdot 31^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01339995565\)
\(L(\frac12)\) \(\approx\) \(0.01339995565\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \)
31 \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
good2 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
5 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
7 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
11 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
17 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
23 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
29 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
43 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
53 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
59 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
79 \( ( 1 + T )^{8}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
83 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
89 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
97 \( ( 1 + T + T^{2} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
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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

−3.88180345881447827499367236554, −3.67061359410339818842833023615, −3.50261976228617090614988571153, −3.33957232403423178344577872674, −3.11084916493848097604344074725, −3.05347511119424936687371012937, −2.98677491332149868115025505495, −2.97512473185519346108129553671, −2.94508247828969019798361984865, −2.53875871296884542977697525632, −2.51718210174376601547771500046, −2.49140244006066644764465428318, −2.41360836780861101953759829356, −2.39494662845543476284909269522, −2.09251135373852284193191385591, −1.89487229226720484518813629903, −1.88539668157069122878263412005, −1.75979620585857502627144965104, −1.54294979728361107838139934850, −1.32663888891945927506916678267, −1.19438386724802193430088843545, −0.976678548477857021595940592447, −0.976301024633020810582688502248, −0.31684410256388412117139953941, −0.05124121472733967101378153411, 0.05124121472733967101378153411, 0.31684410256388412117139953941, 0.976301024633020810582688502248, 0.976678548477857021595940592447, 1.19438386724802193430088843545, 1.32663888891945927506916678267, 1.54294979728361107838139934850, 1.75979620585857502627144965104, 1.88539668157069122878263412005, 1.89487229226720484518813629903, 2.09251135373852284193191385591, 2.39494662845543476284909269522, 2.41360836780861101953759829356, 2.49140244006066644764465428318, 2.51718210174376601547771500046, 2.53875871296884542977697525632, 2.94508247828969019798361984865, 2.97512473185519346108129553671, 2.98677491332149868115025505495, 3.05347511119424936687371012937, 3.11084916493848097604344074725, 3.33957232403423178344577872674, 3.50261976228617090614988571153, 3.67061359410339818842833023615, 3.88180345881447827499367236554

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

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