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Properties

Label 16-423e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.025\times 10^{21}$
Sign $1$
Analytic cond. $3.94439\times 10^{-6}$
Root an. cond. $0.459461$
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  + 2·2-s − 2·3-s + 3·4-s − 4·6-s − 7-s + 2·8-s + 9-s − 6·12-s − 2·14-s + 16-s + 2·17-s + 2·18-s + 2·21-s − 4·24-s − 4·25-s − 3·28-s − 2·32-s + 4·34-s + 3·36-s + 2·37-s + 4·42-s − 4·47-s − 2·48-s − 8·50-s − 4·51-s + 2·53-s − 2·56-s + ⋯
L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s − 4·6-s − 7-s + 2·8-s + 9-s − 6·12-s − 2·14-s + 16-s + 2·17-s + 2·18-s + 2·21-s − 4·24-s − 4·25-s − 3·28-s − 2·32-s + 4·34-s + 3·36-s + 2·37-s + 4·42-s − 4·47-s − 2·48-s − 8·50-s − 4·51-s + 2·53-s − 2·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 47^{8}\)
Sign: $1$
Analytic conductor: \(3.94439\times 10^{-6}\)
Root analytic conductor: \(0.459461\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 47^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2794283636\)
\(L(\frac12)\) \(\approx\) \(0.2794283636\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−5.35290046575934266941915186013, −5.03985278215547249185162285982, −5.02878368975889317506954929294, −4.87368303731495029081758209312, −4.83446835446314400678472897808, −4.79825810786855805095326739351, −4.25875436907266130233692872818, −4.03714651761181730073059658024, −3.97982168942489421517439082962, −3.92641758329400160316985149334, −3.91760896990749318742992186360, −3.50874694957051403424191154082, −3.48954466008293303714467466060, −3.40713398305892743161237063158, −3.19460294627564169633014206233, −3.09669115543710024576826660933, −2.83852832426714649569912142394, −2.70294911625623485015209604010, −2.38065710463709234217052358424, −2.31391733557936781195930898767, −2.10016892991935633818616321260, −1.73547466048131857853209190585, −1.66319743137274956418888759876, −1.41041753636882314956627302363, −0.829276598152766612275609290981, 0.829276598152766612275609290981, 1.41041753636882314956627302363, 1.66319743137274956418888759876, 1.73547466048131857853209190585, 2.10016892991935633818616321260, 2.31391733557936781195930898767, 2.38065710463709234217052358424, 2.70294911625623485015209604010, 2.83852832426714649569912142394, 3.09669115543710024576826660933, 3.19460294627564169633014206233, 3.40713398305892743161237063158, 3.48954466008293303714467466060, 3.50874694957051403424191154082, 3.91760896990749318742992186360, 3.92641758329400160316985149334, 3.97982168942489421517439082962, 4.03714651761181730073059658024, 4.25875436907266130233692872818, 4.79825810786855805095326739351, 4.83446835446314400678472897808, 4.87368303731495029081758209312, 5.02878368975889317506954929294, 5.03985278215547249185162285982, 5.35290046575934266941915186013

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

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