Properties

Degree 16
Conductor $ 3^{8} \cdot 5^{8} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s − 16·7-s + 18·9-s + 68·13-s + 79·16-s + 96·21-s − 110·25-s + 198·27-s + 616·31-s − 1.15e3·37-s − 408·39-s + 548·43-s − 474·48-s + 128·49-s + 16·61-s − 288·63-s + 404·67-s − 2.51e3·73-s + 660·75-s − 1.27e3·81-s − 1.08e3·91-s − 3.69e3·93-s + 1.90e3·97-s − 6.83e3·103-s + 6.93e3·111-s − 1.26e3·112-s + 1.22e3·117-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.863·7-s + 2/3·9-s + 1.45·13-s + 1.23·16-s + 0.997·21-s − 0.879·25-s + 1.41·27-s + 3.56·31-s − 5.13·37-s − 1.67·39-s + 1.94·43-s − 1.42·48-s + 0.373·49-s + 0.0335·61-s − 0.575·63-s + 0.736·67-s − 4.02·73-s + 1.01·75-s − 1.75·81-s − 1.25·91-s − 4.12·93-s + 1.99·97-s − 6.53·103-s + 5.93·111-s − 1.06·112-s + 0.967·117-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(3^{8} \cdot 5^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{15} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 3^{8} \cdot 5^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )$
$L(2)$  $\approx$  $0.602084$
$L(\frac12)$  $\approx$  $0.602084$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5\}$, \(F_p\) is a polynomial of degree 16. If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 15.
$p$$F_p$
bad3 \( 1 + 2 p T + 2 p^{2} T^{2} - 22 p^{2} T^{3} - 158 p^{2} T^{4} - 22 p^{5} T^{5} + 2 p^{8} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \)
5 \( 1 + 22 p T^{2} + 42 p^{3} T^{4} + 22 p^{7} T^{6} + p^{12} T^{8} \)
good2 \( 1 - 79 T^{4} + 39 p^{6} T^{8} - 79 p^{12} T^{12} + p^{24} T^{16} \)
7 \( ( 1 + 8 T + 32 T^{2} + 744 T^{3} - 45202 T^{4} + 744 p^{3} T^{5} + 32 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
11 \( ( 1 - 3334 T^{2} + 6292986 T^{4} - 3334 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
13 \( ( 1 - 34 T + 578 T^{2} - 77418 T^{3} + 10363058 T^{4} - 77418 p^{3} T^{5} + 578 p^{6} T^{6} - 34 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
17 \( 1 - 10588864 T^{4} - 447530075229954 T^{8} - 10588864 p^{12} T^{12} + p^{24} T^{16} \)
19 \( ( 1 - 24664 T^{2} + 245125086 T^{4} - 24664 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
23 \( 1 + 232150736 T^{4} + 54876630002867166 T^{8} + 232150736 p^{12} T^{12} + p^{24} T^{16} \)
29 \( ( 1 + 79166 T^{2} + 2712313506 T^{4} + 79166 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
31 \( ( 1 - 154 T + 36486 T^{2} - 154 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
37 \( ( 1 + 578 T + 167042 T^{2} + 53079474 T^{3} + 15170807378 T^{4} + 53079474 p^{3} T^{5} + 167042 p^{6} T^{6} + 578 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
41 \( ( 1 - 115444 T^{2} + 7614039366 T^{4} - 115444 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
43 \( ( 1 - 274 T + 37538 T^{2} - 17976318 T^{3} + 8415346898 T^{4} - 17976318 p^{3} T^{5} + 37538 p^{6} T^{6} - 274 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
47 \( 1 + 29984206736 T^{4} + \)\(42\!\cdots\!06\)\( T^{8} + 29984206736 p^{12} T^{12} + p^{24} T^{16} \)
53 \( 1 - 24371904064 T^{4} + \)\(31\!\cdots\!06\)\( T^{8} - 24371904064 p^{12} T^{12} + p^{24} T^{16} \)
59 \( ( 1 + 112106 T^{2} + 56049165066 T^{4} + 112106 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
61 \( ( 1 - 2 T + p^{3} T^{2} )^{8} \)
67 \( ( 1 - 202 T + 20402 T^{2} - 3297246 T^{3} - 80373232942 T^{4} - 3297246 p^{3} T^{5} + 20402 p^{6} T^{6} - 202 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
71 \( ( 1 - 863584 T^{2} + 377860054206 T^{4} - 863584 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
73 \( ( 1 + 1256 T + 788768 T^{2} + 733362072 T^{3} + 643873740638 T^{4} + 733362072 p^{3} T^{5} + 788768 p^{6} T^{6} + 1256 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
79 \( ( 1 - 1624084 T^{2} + 1116095863206 T^{4} - 1624084 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
83 \( 1 + 1005705244496 T^{4} + \)\(46\!\cdots\!26\)\( T^{8} + 1005705244496 p^{12} T^{12} + p^{24} T^{16} \)
89 \( ( 1 - 806584 T^{2} + 1118488022286 T^{4} - 806584 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
97 \( ( 1 - 952 T + 453152 T^{2} - 121725576 T^{3} - 583228842562 T^{4} - 121725576 p^{3} T^{5} + 453152 p^{6} T^{6} - 952 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.832573702229850873819797035896, −9.234104993362180615990450167760, −9.070432271025360323698373940548, −8.688393218705620137484404251792, −8.643475881773764191144968347928, −8.576976398107080362336417453962, −8.243528307960224561706635465053, −7.932727633344145449525832173730, −7.64039622096538033022724280243, −7.30324020632429952807597508995, −7.10783018415376297378217035178, −6.74591623147387126665616869644, −6.50255990905518713887888240877, −6.44580005751450636991095555443, −6.05628949291157793786288403308, −5.75433112901286944091839162579, −5.56961244628283330065767177089, −5.29863128763724854429367332798, −4.69659936412278108134805905703, −4.67402148160883643636680785694, −3.90055542462926303190335709383, −3.81923699294316135353858380830, −3.12561044923699290414261651363, −2.82304810281741027661711502846, −1.33574749407580803200854942549, 1.33574749407580803200854942549, 2.82304810281741027661711502846, 3.12561044923699290414261651363, 3.81923699294316135353858380830, 3.90055542462926303190335709383, 4.67402148160883643636680785694, 4.69659936412278108134805905703, 5.29863128763724854429367332798, 5.56961244628283330065767177089, 5.75433112901286944091839162579, 6.05628949291157793786288403308, 6.44580005751450636991095555443, 6.50255990905518713887888240877, 6.74591623147387126665616869644, 7.10783018415376297378217035178, 7.30324020632429952807597508995, 7.64039622096538033022724280243, 7.932727633344145449525832173730, 8.243528307960224561706635465053, 8.576976398107080362336417453962, 8.643475881773764191144968347928, 8.688393218705620137484404251792, 9.070432271025360323698373940548, 9.234104993362180615990450167760, 9.832573702229850873819797035896

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

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