Properties

Degree 16
Conductor $ 2^{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  − 2·2-s + 3·3-s + 4-s + 5-s − 6·6-s + 2·7-s + 9-s − 2·10-s − 13·11-s + 3·12-s − 4·14-s + 3·15-s − 2·17-s − 2·18-s − 3·19-s + 20-s + 6·21-s + 26·22-s + 3·23-s + 11·25-s − 16·27-s + 2·28-s + 11·29-s − 6·30-s + 11·31-s + 2·32-s − 39·33-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.73·3-s + 1/2·4-s + 0.447·5-s − 2.44·6-s + 0.755·7-s + 1/3·9-s − 0.632·10-s − 3.91·11-s + 0.866·12-s − 1.06·14-s + 0.774·15-s − 0.485·17-s − 0.471·18-s − 0.688·19-s + 0.223·20-s + 1.30·21-s + 5.54·22-s + 0.625·23-s + 11/5·25-s − 3.07·27-s + 0.377·28-s + 2.04·29-s − 1.09·30-s + 1.97·31-s + 0.353·32-s − 6.78·33-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 31^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{62} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.257429$
$L(\frac12)$  $\approx$  $0.257429$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;31\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( 1 - 11 T + 60 T^{2} + 11 T^{3} - 991 T^{4} + 11 p T^{5} + 60 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
good3 \( 1 - p T + 8 T^{2} - 5 T^{3} + 28 T^{5} + 8 p T^{6} - 110 T^{7} + 409 T^{8} - 110 p T^{9} + 8 p^{3} T^{10} + 28 p^{3} T^{11} - 5 p^{5} T^{13} + 8 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \)
5 \( 1 - T - 2 p T^{2} + 21 T^{3} + 44 T^{4} - 134 T^{5} + 11 p T^{6} + 384 T^{7} - 959 T^{8} + 384 p T^{9} + 11 p^{3} T^{10} - 134 p^{3} T^{11} + 44 p^{4} T^{12} + 21 p^{5} T^{13} - 2 p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
7 \( 1 - 2 T + 17 T^{2} - 29 T^{3} + 178 T^{4} - 281 T^{5} + 1444 T^{6} - 1692 T^{7} + 10263 T^{8} - 1692 p T^{9} + 1444 p^{2} T^{10} - 281 p^{3} T^{11} + 178 p^{4} T^{12} - 29 p^{5} T^{13} + 17 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 + 13 T + 111 T^{2} + 702 T^{3} + 346 p T^{4} + 17889 T^{5} + 76094 T^{6} + 291046 T^{7} + 1016667 T^{8} + 291046 p T^{9} + 76094 p^{2} T^{10} + 17889 p^{3} T^{11} + 346 p^{5} T^{12} + 702 p^{5} T^{13} + 111 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 - 27 T^{2} - 50 T^{3} + 480 T^{4} + 1000 T^{5} - 4157 T^{6} - 9000 T^{7} + 47779 T^{8} - 9000 p T^{9} - 4157 p^{2} T^{10} + 1000 p^{3} T^{11} + 480 p^{4} T^{12} - 50 p^{5} T^{13} - 27 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 + 2 T - 43 T^{2} - 3 p T^{3} + 328 T^{4} - 759 T^{5} + 12554 T^{6} + 17852 T^{7} - 360647 T^{8} + 17852 p T^{9} + 12554 p^{2} T^{10} - 759 p^{3} T^{11} + 328 p^{4} T^{12} - 3 p^{6} T^{13} - 43 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 + 3 T - 26 T^{2} + 9 T^{3} + 762 T^{4} - 582 T^{5} - 13702 T^{6} - 3486 T^{7} + 152081 T^{8} - 3486 p T^{9} - 13702 p^{2} T^{10} - 582 p^{3} T^{11} + 762 p^{4} T^{12} + 9 p^{5} T^{13} - 26 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 3 T + 7 T^{2} - 204 T^{3} + 1161 T^{4} - 3168 T^{5} + 22910 T^{6} - 131367 T^{7} + 625349 T^{8} - 131367 p T^{9} + 22910 p^{2} T^{10} - 3168 p^{3} T^{11} + 1161 p^{4} T^{12} - 204 p^{5} T^{13} + 7 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 11 T + 14 T^{2} + 74 T^{3} + 506 T^{4} - 1933 T^{5} + 11316 T^{6} - 73148 T^{7} + 5907 T^{8} - 73148 p T^{9} + 11316 p^{2} T^{10} - 1933 p^{3} T^{11} + 506 p^{4} T^{12} + 74 p^{5} T^{13} + 14 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 - 6 T - 78 T^{2} + 330 T^{3} + 4225 T^{4} - 7809 T^{5} - 196479 T^{6} + 132135 T^{7} + 7291959 T^{8} + 132135 p T^{9} - 196479 p^{2} T^{10} - 7809 p^{3} T^{11} + 4225 p^{4} T^{12} + 330 p^{5} T^{13} - 78 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 + 19 T + 266 T^{2} + 2343 T^{3} + 19987 T^{4} + 136629 T^{5} + 992162 T^{6} + 140873 p T^{7} + 39291616 T^{8} + 140873 p^{2} T^{9} + 992162 p^{2} T^{10} + 136629 p^{3} T^{11} + 19987 p^{4} T^{12} + 2343 p^{5} T^{13} + 266 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 + 26 T + 283 T^{2} + 1288 T^{3} - 1732 T^{4} - 36097 T^{5} + 136466 T^{6} + 4437349 T^{7} + 39368833 T^{8} + 4437349 p T^{9} + 136466 p^{2} T^{10} - 36097 p^{3} T^{11} - 1732 p^{4} T^{12} + 1288 p^{5} T^{13} + 283 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 - T - 22 T^{2} - 137 T^{3} + 2366 T^{4} + 4924 T^{5} - 33810 T^{6} - 202576 T^{7} + 5800929 T^{8} - 202576 p T^{9} - 33810 p^{2} T^{10} + 4924 p^{3} T^{11} + 2366 p^{4} T^{12} - 137 p^{5} T^{13} - 22 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 17 T + 153 T^{2} - 1080 T^{3} + 5585 T^{4} - 1443 T^{5} - 181831 T^{6} + 2916400 T^{7} - 28141191 T^{8} + 2916400 p T^{9} - 181831 p^{2} T^{10} - 1443 p^{3} T^{11} + 5585 p^{4} T^{12} - 1080 p^{5} T^{13} + 153 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 8 T + 44 T^{2} - 949 T^{3} + 10412 T^{4} - 60868 T^{5} + 591603 T^{6} - 5355299 T^{7} + 38621781 T^{8} - 5355299 p T^{9} + 591603 p^{2} T^{10} - 60868 p^{3} T^{11} + 10412 p^{4} T^{12} - 949 p^{5} T^{13} + 44 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
61 \( ( 1 - 24 T + 385 T^{2} - 4356 T^{3} + 39249 T^{4} - 4356 p T^{5} + 385 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 12 T - 138 T^{2} + 1146 T^{3} + 23563 T^{4} - 99741 T^{5} - 2275281 T^{6} + 1600923 T^{7} + 194106273 T^{8} + 1600923 p T^{9} - 2275281 p^{2} T^{10} - 99741 p^{3} T^{11} + 23563 p^{4} T^{12} + 1146 p^{5} T^{13} - 138 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 9 T + 146 T^{2} - 1158 T^{3} + 10347 T^{4} - 88539 T^{5} + 812197 T^{6} - 7118208 T^{7} + 60753011 T^{8} - 7118208 p T^{9} + 812197 p^{2} T^{10} - 88539 p^{3} T^{11} + 10347 p^{4} T^{12} - 1158 p^{5} T^{13} + 146 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 33 T + 768 T^{2} + 12580 T^{3} + 176025 T^{4} + 2064592 T^{5} + 22155244 T^{6} + 211797255 T^{7} + 1902972349 T^{8} + 211797255 p T^{9} + 22155244 p^{2} T^{10} + 2064592 p^{3} T^{11} + 176025 p^{4} T^{12} + 12580 p^{5} T^{13} + 768 p^{6} T^{14} + 33 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 + 19 T + 394 T^{2} + 5504 T^{3} + 899 p T^{4} + 825307 T^{5} + 8678231 T^{6} + 88869362 T^{7} + 794929807 T^{8} + 88869362 p T^{9} + 8678231 p^{2} T^{10} + 825307 p^{3} T^{11} + 899 p^{5} T^{12} + 5504 p^{5} T^{13} + 394 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 - 40 T + 698 T^{2} - 6665 T^{3} + 27170 T^{4} + 192160 T^{5} - 3936477 T^{6} + 31340435 T^{7} - 222267711 T^{8} + 31340435 p T^{9} - 3936477 p^{2} T^{10} + 192160 p^{3} T^{11} + 27170 p^{4} T^{12} - 6665 p^{5} T^{13} + 698 p^{6} T^{14} - 40 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 8 T - 145 T^{2} + 1605 T^{3} + 12835 T^{4} - 42189 T^{5} - 2038663 T^{6} - 1693160 T^{7} + 286639345 T^{8} - 1693160 p T^{9} - 2038663 p^{2} T^{10} - 42189 p^{3} T^{11} + 12835 p^{4} T^{12} + 1605 p^{5} T^{13} - 145 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 23 T + 244 T^{2} + 3433 T^{3} + 48488 T^{4} + 513248 T^{5} + 6282506 T^{6} + 62989744 T^{7} + 526428043 T^{8} + 62989744 p T^{9} + 6282506 p^{2} T^{10} + 513248 p^{3} T^{11} + 48488 p^{4} T^{12} + 3433 p^{5} T^{13} + 244 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \)
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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

−7.35275923943755253568160755527, −7.01422586599828320761250609457, −6.90613850966011603735014023399, −6.87828920332265777958628356260, −6.77844516328932417724933988781, −6.54468595342712254870122971035, −6.37092260269743223941570125498, −6.10544036322433664435624939630, −5.44499344304181360403738367522, −5.38473088977579549892485904416, −5.35222001588186308591548519493, −5.27991098441012014176084387204, −5.06947911473599281462363766351, −5.05546304975654657425411313163, −4.35427544419291766959026278234, −4.34384084681549357606701331316, −4.23654806967432294815915939694, −3.52266908413777707220955037555, −3.29481454569935208688558704031, −3.27308719821372791226150532040, −2.66872410555293996329814660535, −2.59568837580838752382092854010, −2.49107696846506670613093101132, −2.29641160850963261264562225980, −1.53072920549421909342227750867, 1.53072920549421909342227750867, 2.29641160850963261264562225980, 2.49107696846506670613093101132, 2.59568837580838752382092854010, 2.66872410555293996329814660535, 3.27308719821372791226150532040, 3.29481454569935208688558704031, 3.52266908413777707220955037555, 4.23654806967432294815915939694, 4.34384084681549357606701331316, 4.35427544419291766959026278234, 5.05546304975654657425411313163, 5.06947911473599281462363766351, 5.27991098441012014176084387204, 5.35222001588186308591548519493, 5.38473088977579549892485904416, 5.44499344304181360403738367522, 6.10544036322433664435624939630, 6.37092260269743223941570125498, 6.54468595342712254870122971035, 6.77844516328932417724933988781, 6.87828920332265777958628356260, 6.90613850966011603735014023399, 7.01422586599828320761250609457, 7.35275923943755253568160755527

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

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