Properties

Label 16-1400e8-1.1-c0e8-0-1
Degree $16$
Conductor $1.476\times 10^{25}$
Sign $1$
Analytic cond. $0.0567912$
Root an. cond. $0.835877$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·11-s + 16-s + 2·81-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 4·11-s + 16-s + 2·81-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(0.0567912\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5397276320\)
\(L(\frac12)\) \(\approx\) \(0.5397276320\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
5 \( 1 \)
7 \( ( 1 + T^{4} )^{2} \)
good3 \( ( 1 - T^{4} + T^{8} )^{2} \)
11 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
13 \( ( 1 - T^{4} + T^{8} )^{2} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
23 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
29 \( ( 1 + T^{2} )^{8} \)
31 \( ( 1 - T^{2} + T^{4} )^{4} \)
37 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
41 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
53 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
59 \( ( 1 - T^{2} + T^{4} )^{4} \)
61 \( ( 1 - T^{2} + T^{4} )^{4} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 - T )^{8}( 1 + T )^{8} \)
73 \( ( 1 - T^{4} + T^{8} )^{2} \)
79 \( ( 1 - T^{2} + T^{4} )^{4} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 - T^{2} + T^{4} )^{4} \)
97 \( ( 1 + T^{4} )^{4} \)
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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

−4.42169588783792847640414017993, −4.06130047328817027678320173259, −3.96842703862419395729650348543, −3.79061083581996722743594145270, −3.77007004961947542021388278941, −3.76622246855577697835317856075, −3.59638645724153467551674027748, −3.38833353448210888562074158099, −3.20376541789548576100150409441, −3.06182138894486125860872480624, −2.94604232055429392529926994202, −2.94433072391835543899915454449, −2.59671748629072887552206970704, −2.53544836201763875529287668594, −2.45745800860247170163570861473, −2.44921292152753361691213962454, −2.29848189773770337800704330594, −2.07627177707208632520451503859, −1.79225768456469176711182296686, −1.63034855605937855165070787651, −1.49096835207782934262291208101, −1.25999334972318086574075162043, −1.19156858487768440120641321874, −0.63045122520183065317989600293, −0.49481425394406403515591099010, 0.49481425394406403515591099010, 0.63045122520183065317989600293, 1.19156858487768440120641321874, 1.25999334972318086574075162043, 1.49096835207782934262291208101, 1.63034855605937855165070787651, 1.79225768456469176711182296686, 2.07627177707208632520451503859, 2.29848189773770337800704330594, 2.44921292152753361691213962454, 2.45745800860247170163570861473, 2.53544836201763875529287668594, 2.59671748629072887552206970704, 2.94433072391835543899915454449, 2.94604232055429392529926994202, 3.06182138894486125860872480624, 3.20376541789548576100150409441, 3.38833353448210888562074158099, 3.59638645724153467551674027748, 3.76622246855577697835317856075, 3.77007004961947542021388278941, 3.79061083581996722743594145270, 3.96842703862419395729650348543, 4.06130047328817027678320173259, 4.42169588783792847640414017993

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

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