Properties

Label 16-160e8-1.1-c3e8-0-2
Degree $16$
Conductor $4.295\times 10^{17}$
Sign $1$
Analytic cond. $6.30795\times 10^{7}$
Root an. cond. $3.07250$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·5-s + 32·9-s + 308·25-s + 624·29-s − 192·41-s + 1.02e3·45-s + 160·49-s + 2.11e3·61-s − 764·81-s + 80·89-s − 3.53e3·101-s + 3.39e3·109-s − 664·121-s − 1.69e3·125-s + 127-s + 131-s + 137-s + 139-s + 1.99e4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.41e3·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2.86·5-s + 1.18·9-s + 2.46·25-s + 3.99·29-s − 0.731·41-s + 3.39·45-s + 0.466·49-s + 4.43·61-s − 1.04·81-s + 0.0952·89-s − 3.48·101-s + 2.98·109-s − 0.498·121-s − 1.21·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 11.4·145-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.46·169-s + 0.000439·173-s + 0.000417·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(6.30795\times 10^{7}\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{160} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 5^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(23.62666310\)
\(L(\frac12)\) \(\approx\) \(23.62666310\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 - 16 T + 46 p T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
good3 \( ( 1 - 16 T^{2} + 766 T^{4} - 16 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
7 \( ( 1 - 80 T^{2} + 212622 T^{4} - 80 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
11 \( ( 1 + 332 T^{2} + 474102 T^{4} + 332 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
13 \( ( 1 - 2708 T^{2} + 7852758 T^{4} - 2708 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
17 \( ( 1 - 4450 T^{2} + p^{6} T^{4} )^{4} \)
19 \( ( 1 - 2644 T^{2} + 87237846 T^{4} - 2644 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
23 \( ( 1 - 36816 T^{2} + 602323406 T^{4} - 36816 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
29 \( ( 1 - 156 T + 33358 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
31 \( ( 1 + 95612 T^{2} + 4010875782 T^{4} + 95612 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
37 \( ( 1 - 160436 T^{2} + 8226342 p^{2} T^{4} - 160436 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
41 \( ( 1 + 48 T + 124222 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
43 \( ( 1 - 66320 T^{2} - 2039763618 T^{4} - 66320 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
47 \( ( 1 - 204240 T^{2} + 24337141742 T^{4} - 204240 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
53 \( ( 1 - 444020 T^{2} + 93573616758 T^{4} - 444020 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
59 \( ( 1 + 507788 T^{2} + 128506595382 T^{4} + 507788 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
61 \( ( 1 - 528 T + 462422 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
67 \( ( 1 - 814480 T^{2} + 336078840702 T^{4} - 814480 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
71 \( ( 1 + 1385052 T^{2} + 735523082342 T^{4} + 1385052 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
73 \( ( 1 - 971620 T^{2} + 454375527078 T^{4} - 971620 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
79 \( ( 1 + 756668 T^{2} + 456844629702 T^{4} + 756668 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
83 \( ( 1 - 300176 T^{2} + 429308372286 T^{4} - 300176 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
89 \( ( 1 - 20 T + 872438 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
97 \( ( 1 - 534850 T^{2} + p^{6} T^{4} )^{4} \)
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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

−5.30427269549830813412618538750, −5.29429227994771338051606049269, −5.23731802382952637500698176788, −4.91649478382524877285923176531, −4.68967609455711988711723475681, −4.64008319903868148717623633521, −4.38488680841972823627493930141, −4.35992245518714077008759898351, −3.92580112744876274726015212916, −3.79901246545787289774327788112, −3.76621696162690482758448157665, −3.55167686017964275347178401325, −3.08165116074695062012308450986, −2.84104206802738002734833689816, −2.81761714331742832738234726732, −2.54259038736404796202911993060, −2.48600369849055589303356204426, −2.11307239092703246920526119746, −1.86548085766597083425000003769, −1.72246242498007187356295720300, −1.56193145737809749519729641611, −1.23966192492169492694470605436, −0.847848186583845727473206985292, −0.74177337154159347003225054451, −0.40005941588809582139455162648, 0.40005941588809582139455162648, 0.74177337154159347003225054451, 0.847848186583845727473206985292, 1.23966192492169492694470605436, 1.56193145737809749519729641611, 1.72246242498007187356295720300, 1.86548085766597083425000003769, 2.11307239092703246920526119746, 2.48600369849055589303356204426, 2.54259038736404796202911993060, 2.81761714331742832738234726732, 2.84104206802738002734833689816, 3.08165116074695062012308450986, 3.55167686017964275347178401325, 3.76621696162690482758448157665, 3.79901246545787289774327788112, 3.92580112744876274726015212916, 4.35992245518714077008759898351, 4.38488680841972823627493930141, 4.64008319903868148717623633521, 4.68967609455711988711723475681, 4.91649478382524877285923176531, 5.23731802382952637500698176788, 5.29429227994771338051606049269, 5.30427269549830813412618538750

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

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