Properties

Degree 16
Conductor $ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 6·4-s + 24·11-s − 16·13-s − 15·16-s − 24·17-s − 96·22-s + 8·23-s + 64·26-s + 24·32-s + 96·34-s + 32·41-s + 144·44-s − 32·46-s − 12·47-s − 96·52-s − 4·53-s + 24·59-s − 6·64-s − 48·67-s − 144·68-s + 24·73-s + 24·79-s − 128·82-s + 16·89-s + 48·92-s + 48·94-s + ⋯
L(s)  = 1  − 2.82·2-s + 3·4-s + 7.23·11-s − 4.43·13-s − 3.75·16-s − 5.82·17-s − 20.4·22-s + 1.66·23-s + 12.5·26-s + 4.24·32-s + 16.4·34-s + 4.99·41-s + 21.7·44-s − 4.71·46-s − 1.75·47-s − 13.3·52-s − 0.549·53-s + 3.12·59-s − 3/4·64-s − 5.86·67-s − 17.4·68-s + 2.80·73-s + 2.70·79-s − 14.1·82-s + 1.69·89-s + 5.00·92-s + 4.95·94-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.1157034823$
$L(\frac12)$  $\approx$  $0.1157034823$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{4} \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - 94 T^{4} + p^{4} T^{8} \)
good11 \( 1 - 24 T + 304 T^{2} - 2688 T^{3} + 18481 T^{4} - 104232 T^{5} + 496816 T^{6} - 2036016 T^{7} + 7239280 T^{8} - 2036016 p T^{9} + 496816 p^{2} T^{10} - 104232 p^{3} T^{11} + 18481 p^{4} T^{12} - 2688 p^{5} T^{13} + 304 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
13 \( ( 1 + 8 T + 72 T^{2} + 328 T^{3} + 1535 T^{4} + 328 p T^{5} + 72 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( 1 + 24 T + 316 T^{2} + 2976 T^{3} + 22330 T^{4} + 141240 T^{5} + 776752 T^{6} + 3781944 T^{7} + 16463059 T^{8} + 3781944 p T^{9} + 776752 p^{2} T^{10} + 141240 p^{3} T^{11} + 22330 p^{4} T^{12} + 2976 p^{5} T^{13} + 316 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 + 58 T^{2} + 99 p T^{4} + 1872 T^{5} + 44906 T^{6} + 71184 T^{7} + 869156 T^{8} + 71184 p T^{9} + 44906 p^{2} T^{10} + 1872 p^{3} T^{11} + 99 p^{5} T^{12} + 58 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 - 4 T - 31 T^{2} - 4 T^{3} + 1312 T^{4} - 4 p T^{5} - 31 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 36 T^{2} + 470 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( 1 + 100 T^{2} + 5706 T^{4} + 237200 T^{6} + 7904915 T^{8} + 237200 p^{2} T^{10} + 5706 p^{4} T^{12} + 100 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 + 72 T^{2} + 1441 T^{4} - 2520 T^{5} + 74088 T^{6} - 269496 T^{7} + 5102928 T^{8} - 269496 p T^{9} + 74088 p^{2} T^{10} - 2520 p^{3} T^{11} + 1441 p^{4} T^{12} + 72 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 - 16 T + 232 T^{2} - 2000 T^{3} + 15591 T^{4} - 2000 p T^{5} + 232 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 296 T^{2} + 39900 T^{4} - 3207640 T^{6} + 168538406 T^{8} - 3207640 p^{2} T^{10} + 39900 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 + 12 T + 182 T^{2} + 1608 T^{3} + 14001 T^{4} + 79344 T^{5} + 532582 T^{6} + 2135988 T^{7} + 16324772 T^{8} + 2135988 p T^{9} + 532582 p^{2} T^{10} + 79344 p^{3} T^{11} + 14001 p^{4} T^{12} + 1608 p^{5} T^{13} + 182 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 4 T - 66 T^{2} - 984 T^{3} - 631 T^{4} + 50712 T^{5} + 363998 T^{6} - 1081556 T^{7} - 22439580 T^{8} - 1081556 p T^{9} + 363998 p^{2} T^{10} + 50712 p^{3} T^{11} - 631 p^{4} T^{12} - 984 p^{5} T^{13} - 66 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 24 T + 224 T^{2} - 1392 T^{3} + 9358 T^{4} - 16056 T^{5} - 613888 T^{6} + 6806952 T^{7} - 48734957 T^{8} + 6806952 p T^{9} - 613888 p^{2} T^{10} - 16056 p^{3} T^{11} + 9358 p^{4} T^{12} - 1392 p^{5} T^{13} + 224 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 124 T^{2} + 6186 T^{4} + 30240 T^{5} + 293552 T^{6} + 4411200 T^{7} + 18080963 T^{8} + 4411200 p T^{9} + 293552 p^{2} T^{10} + 30240 p^{3} T^{11} + 6186 p^{4} T^{12} + 124 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 + 48 T + 1108 T^{2} + 16320 T^{3} + 166858 T^{4} + 1140048 T^{5} + 3573520 T^{6} - 22726896 T^{7} - 369529661 T^{8} - 22726896 p T^{9} + 3573520 p^{2} T^{10} + 1140048 p^{3} T^{11} + 166858 p^{4} T^{12} + 16320 p^{5} T^{13} + 1108 p^{6} T^{14} + 48 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 4371368 p^{2} T^{10} + 41532 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} \)
73 \( ( 1 - 12 T - 6 T^{2} - 48 T^{3} + 6659 T^{4} - 48 p T^{5} - 6 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 - 24 T + 252 T^{2} - 1584 T^{3} + 5050 T^{4} - 17832 T^{5} - 542160 T^{6} + 18801768 T^{7} - 226869549 T^{8} + 18801768 p T^{9} - 542160 p^{2} T^{10} - 17832 p^{3} T^{11} + 5050 p^{4} T^{12} - 1584 p^{5} T^{13} + 252 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 - 232 T^{2} + 39708 T^{4} - 4492952 T^{6} + 427617638 T^{8} - 4492952 p^{2} T^{10} + 39708 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \)
89 \( 1 - 16 T - 48 T^{2} + 1824 T^{3} + 2078 T^{4} - 135504 T^{5} + 38912 T^{6} + 7970384 T^{7} - 72309597 T^{8} + 7970384 p T^{9} + 38912 p^{2} T^{10} - 135504 p^{3} T^{11} + 2078 p^{4} T^{12} + 1824 p^{5} T^{13} - 48 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
97 \( ( 1 - 24 T + 276 T^{2} - 1608 T^{3} + 10022 T^{4} - 1608 p T^{5} + 276 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−3.70419270360435386094816291168, −3.45880484291004221923772014686, −3.39154013557628853924873472540, −3.28886241479858530648880856156, −2.94325682858416534763234846787, −2.92375572000627683572015193297, −2.81239791582229750266283543143, −2.59321066645864603867200225054, −2.53000103722909387761348555787, −2.37381928684835693290648921585, −2.32249445507596185374696160750, −2.10445356163746052383419909438, −1.95823042362387622248127077116, −1.94049754630946054162080584897, −1.85200370240362113109299328282, −1.61849029001068559197464955934, −1.59491326049590042575070228288, −1.29726098571147490906028719398, −1.17679016284614833623606202881, −1.10186943764415264914703874207, −0.76869102993845650164557105187, −0.64819670111535486085429715790, −0.58995651033237408201130671946, −0.46644214679692858849003423731, −0.05512405943901193263823027443, 0.05512405943901193263823027443, 0.46644214679692858849003423731, 0.58995651033237408201130671946, 0.64819670111535486085429715790, 0.76869102993845650164557105187, 1.10186943764415264914703874207, 1.17679016284614833623606202881, 1.29726098571147490906028719398, 1.59491326049590042575070228288, 1.61849029001068559197464955934, 1.85200370240362113109299328282, 1.94049754630946054162080584897, 1.95823042362387622248127077116, 2.10445356163746052383419909438, 2.32249445507596185374696160750, 2.37381928684835693290648921585, 2.53000103722909387761348555787, 2.59321066645864603867200225054, 2.81239791582229750266283543143, 2.92375572000627683572015193297, 2.94325682858416534763234846787, 3.28886241479858530648880856156, 3.39154013557628853924873472540, 3.45880484291004221923772014686, 3.70419270360435386094816291168

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

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