Properties

Degree 16
Conductor $ 2^{8} \cdot 5^{8} \cdot 13^{8} \cdot 31^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 3-s + 36·4-s − 8·5-s − 8·6-s + 7-s + 120·8-s − 7·9-s − 64·10-s + 4·11-s − 36·12-s − 8·13-s + 8·14-s + 8·15-s + 330·16-s − 5·17-s − 56·18-s + 2·19-s − 288·20-s − 21-s + 32·22-s + 4·23-s − 120·24-s + 36·25-s − 64·26-s + 12·27-s + 36·28-s + ⋯
L(s)  = 1  + 5.65·2-s − 0.577·3-s + 18·4-s − 3.57·5-s − 3.26·6-s + 0.377·7-s + 42.4·8-s − 7/3·9-s − 20.2·10-s + 1.20·11-s − 10.3·12-s − 2.21·13-s + 2.13·14-s + 2.06·15-s + 82.5·16-s − 1.21·17-s − 13.1·18-s + 0.458·19-s − 64.3·20-s − 0.218·21-s + 6.82·22-s + 0.834·23-s − 24.4·24-s + 36/5·25-s − 12.5·26-s + 2.30·27-s + 6.80·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 5^{8} \cdot 13^{8} \cdot 31^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4030} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{8} \cdot 5^{8} \cdot 13^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $651.6349274$
$L(\frac12)$  $\approx$  $651.6349274$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13,\;31\}$, \(F_p\) is a polynomial of degree 16. If $p \in \{2,\;5,\;13,\;31\}$, then $F_p$ is a polynomial of degree at most 15.
$p$$F_p$
bad2 \( ( 1 - T )^{8} \)
5 \( ( 1 + T )^{8} \)
13 \( ( 1 + T )^{8} \)
31 \( ( 1 + T )^{8} \)
good3 \( 1 + T + 8 T^{2} + p T^{3} + 28 T^{4} + p T^{5} + 101 T^{6} + 59 T^{7} + 376 T^{8} + 59 p T^{9} + 101 p^{2} T^{10} + p^{4} T^{11} + 28 p^{4} T^{12} + p^{6} T^{13} + 8 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
7 \( 1 - T + 23 T^{2} - 9 T^{3} + 286 T^{4} - 2 T^{5} + 2591 T^{6} + 860 T^{7} + 2719 p T^{8} + 860 p T^{9} + 2591 p^{2} T^{10} - 2 p^{3} T^{11} + 286 p^{4} T^{12} - 9 p^{5} T^{13} + 23 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 - 4 T + 35 T^{2} - 92 T^{3} + 570 T^{4} - 1142 T^{5} + 8460 T^{6} - 18960 T^{7} + 112016 T^{8} - 18960 p T^{9} + 8460 p^{2} T^{10} - 1142 p^{3} T^{11} + 570 p^{4} T^{12} - 92 p^{5} T^{13} + 35 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 + 5 T + 76 T^{2} + 316 T^{3} + 2472 T^{4} + 7890 T^{5} + 48351 T^{6} + 117395 T^{7} + 794833 T^{8} + 117395 p T^{9} + 48351 p^{2} T^{10} + 7890 p^{3} T^{11} + 2472 p^{4} T^{12} + 316 p^{5} T^{13} + 76 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 - 2 T + 127 T^{2} - 212 T^{3} + 7364 T^{4} - 10221 T^{5} + 256697 T^{6} - 295892 T^{7} + 311539 p T^{8} - 295892 p T^{9} + 256697 p^{2} T^{10} - 10221 p^{3} T^{11} + 7364 p^{4} T^{12} - 212 p^{5} T^{13} + 127 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 4 T + 96 T^{2} - 209 T^{3} + 3575 T^{4} + 1055 T^{5} + 61299 T^{6} + 295920 T^{7} + 826129 T^{8} + 295920 p T^{9} + 61299 p^{2} T^{10} + 1055 p^{3} T^{11} + 3575 p^{4} T^{12} - 209 p^{5} T^{13} + 96 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 11 T + 202 T^{2} - 1582 T^{3} + 17300 T^{4} - 108166 T^{5} + 894511 T^{6} - 4650835 T^{7} + 31222257 T^{8} - 4650835 p T^{9} + 894511 p^{2} T^{10} - 108166 p^{3} T^{11} + 17300 p^{4} T^{12} - 1582 p^{5} T^{13} + 202 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 - 19 T + 314 T^{2} - 3110 T^{3} + 29965 T^{4} - 218172 T^{5} + 1709699 T^{6} - 10840954 T^{7} + 73835711 T^{8} - 10840954 p T^{9} + 1709699 p^{2} T^{10} - 218172 p^{3} T^{11} + 29965 p^{4} T^{12} - 3110 p^{5} T^{13} + 314 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 10 T + 91 T^{2} - 49 T^{3} - 68 T^{4} + 17396 T^{5} + 86172 T^{6} - 695635 T^{7} + 12420976 T^{8} - 695635 p T^{9} + 86172 p^{2} T^{10} + 17396 p^{3} T^{11} - 68 p^{4} T^{12} - 49 p^{5} T^{13} + 91 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 19 T + 407 T^{2} - 5371 T^{3} + 66655 T^{4} - 664423 T^{5} + 5970052 T^{6} - 46631725 T^{7} + 324812790 T^{8} - 46631725 p T^{9} + 5970052 p^{2} T^{10} - 664423 p^{3} T^{11} + 66655 p^{4} T^{12} - 5371 p^{5} T^{13} + 407 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 - 11 T + 225 T^{2} - 2225 T^{3} + 24230 T^{4} - 211050 T^{5} + 1747717 T^{6} - 13119002 T^{7} + 94179647 T^{8} - 13119002 p T^{9} + 1747717 p^{2} T^{10} - 211050 p^{3} T^{11} + 24230 p^{4} T^{12} - 2225 p^{5} T^{13} + 225 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 8 T + 207 T^{2} - 1265 T^{3} + 20186 T^{4} - 90780 T^{5} + 1205468 T^{6} - 4219371 T^{7} + 62020892 T^{8} - 4219371 p T^{9} + 1205468 p^{2} T^{10} - 90780 p^{3} T^{11} + 20186 p^{4} T^{12} - 1265 p^{5} T^{13} + 207 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 28 T + 574 T^{2} - 8171 T^{3} + 103007 T^{4} - 18479 p T^{5} + 10828067 T^{6} - 94511782 T^{7} + 774302971 T^{8} - 94511782 p T^{9} + 10828067 p^{2} T^{10} - 18479 p^{4} T^{11} + 103007 p^{4} T^{12} - 8171 p^{5} T^{13} + 574 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 12 T + 269 T^{2} + 2794 T^{3} + 37891 T^{4} + 334749 T^{5} + 3633315 T^{6} + 28248553 T^{7} + 255858551 T^{8} + 28248553 p T^{9} + 3633315 p^{2} T^{10} + 334749 p^{3} T^{11} + 37891 p^{4} T^{12} + 2794 p^{5} T^{13} + 269 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 24 T + 622 T^{2} - 10124 T^{3} + 156090 T^{4} - 1898413 T^{5} + 21550379 T^{6} - 204805961 T^{7} + 1820222164 T^{8} - 204805961 p T^{9} + 21550379 p^{2} T^{10} - 1898413 p^{3} T^{11} + 156090 p^{4} T^{12} - 10124 p^{5} T^{13} + 622 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 18 T + 447 T^{2} - 5042 T^{3} + 76493 T^{4} - 691116 T^{5} + 8646686 T^{6} - 69103856 T^{7} + 728626206 T^{8} - 69103856 p T^{9} + 8646686 p^{2} T^{10} - 691116 p^{3} T^{11} + 76493 p^{4} T^{12} - 5042 p^{5} T^{13} + 447 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 3 T + 459 T^{2} + 849 T^{3} + 97306 T^{4} + 113650 T^{5} + 12677766 T^{6} + 10423224 T^{7} + 1114097216 T^{8} + 10423224 p T^{9} + 12677766 p^{2} T^{10} + 113650 p^{3} T^{11} + 97306 p^{4} T^{12} + 849 p^{5} T^{13} + 459 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 22 T + 369 T^{2} - 4244 T^{3} + 45788 T^{4} - 356636 T^{5} + 2640644 T^{6} - 14140098 T^{7} + 119972720 T^{8} - 14140098 p T^{9} + 2640644 p^{2} T^{10} - 356636 p^{3} T^{11} + 45788 p^{4} T^{12} - 4244 p^{5} T^{13} + 369 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 - 17 T + 463 T^{2} - 6685 T^{3} + 109338 T^{4} - 1285924 T^{5} + 16188075 T^{6} - 157647848 T^{7} + 1615294919 T^{8} - 157647848 p T^{9} + 16188075 p^{2} T^{10} - 1285924 p^{3} T^{11} + 109338 p^{4} T^{12} - 6685 p^{5} T^{13} + 463 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 17 T + 500 T^{2} - 5030 T^{3} + 79730 T^{4} - 341626 T^{5} + 4065435 T^{6} + 24086641 T^{7} + 40035651 T^{8} + 24086641 p T^{9} + 4065435 p^{2} T^{10} - 341626 p^{3} T^{11} + 79730 p^{4} T^{12} - 5030 p^{5} T^{13} + 500 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 24 T + 524 T^{2} + 8234 T^{3} + 117230 T^{4} + 1365788 T^{5} + 15895736 T^{6} + 161697301 T^{7} + 1656569083 T^{8} + 161697301 p T^{9} + 15895736 p^{2} T^{10} + 1365788 p^{3} T^{11} + 117230 p^{4} T^{12} + 8234 p^{5} T^{13} + 524 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
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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

−3.63069521052672956646084222740, −3.23448033319172831544523815564, −3.22320382963540322605273198910, −3.13346303150528607789275852023, −3.11829652638582775914835228166, −3.10269842363745649311820457600, −2.97556825462200228714434827616, −2.89826261053644904956625560025, −2.79817729689008600843588397467, −2.47929094525654821875115895068, −2.43618260399829584679897347904, −2.25596595565586168057176863628, −2.23018497563588620232285605018, −2.21788518626850889625733199442, −2.13345603621137107650233993390, −1.90715147558623358983163568697, −1.84643065643365311680470427485, −1.40669157213978022784888625685, −1.26656015597652495730999352302, −0.861313946858545281333724320516, −0.854785005844326845476791014616, −0.808882383756333773341717806240, −0.68526156478619038567400605932, −0.46173468622084347996312533209, −0.46056767795535108364305203318, 0.46056767795535108364305203318, 0.46173468622084347996312533209, 0.68526156478619038567400605932, 0.808882383756333773341717806240, 0.854785005844326845476791014616, 0.861313946858545281333724320516, 1.26656015597652495730999352302, 1.40669157213978022784888625685, 1.84643065643365311680470427485, 1.90715147558623358983163568697, 2.13345603621137107650233993390, 2.21788518626850889625733199442, 2.23018497563588620232285605018, 2.25596595565586168057176863628, 2.43618260399829584679897347904, 2.47929094525654821875115895068, 2.79817729689008600843588397467, 2.89826261053644904956625560025, 2.97556825462200228714434827616, 3.10269842363745649311820457600, 3.11829652638582775914835228166, 3.13346303150528607789275852023, 3.22320382963540322605273198910, 3.23448033319172831544523815564, 3.63069521052672956646084222740

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

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