Properties

Label 16-84e16-1.1-c1e8-0-1
Degree $16$
Conductor $6.144\times 10^{30}$
Sign $1$
Analytic cond. $1.01551\times 10^{14}$
Root an. cond. $7.50616$
Motivic weight $1$
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  − 32·37-s + 32·43-s + 32·79-s − 32·109-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 5.26·37-s + 4.87·43-s + 3.60·79-s − 3.06·109-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(1.01551\times 10^{14}\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7056} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3300486965\)
\(L(\frac12)\) \(\approx\) \(0.3300486965\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 + 48 T^{4} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 32 T^{2} + 592 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 16 T^{2} + 64 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + 4 T^{2} + 694 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 - 92 T^{2} + 3670 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 44 T^{2} + 2374 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + 96 T^{2} + 4608 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + 108 T^{2} + 7302 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{4} \)
59 \( ( 1 + 156 T^{2} + 13014 T^{4} + 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 176 T^{2} + 14128 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 - 260 T^{2} + 26854 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 240 T^{2} + 24480 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
83 \( ( 1 + 12 T^{2} + 13302 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 192 T^{2} + 23136 T^{4} + 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 336 T^{2} + 46464 T^{4} - 336 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−3.22179582374106985687948057846, −3.19069248639252601088189347410, −3.18939853358993259507778788679, −2.77269385001125963418677357324, −2.72064048450740367774580464262, −2.67260383093966421076349675398, −2.51824631695324832158276186123, −2.47445277991148694205461537381, −2.26776952378465629430712287428, −2.25977559065271480881644531148, −2.12488819413898098326702039720, −2.05120183344966994532669281005, −1.87856591472453537333759649943, −1.86637093752516451751141309969, −1.52751249273918298430174049297, −1.52579825895698779457666885764, −1.29191832746272161862674408278, −1.27567932158929819208969679113, −1.03592345230120291762210919898, −0.936199096923585792646386629966, −0.882724409831705366553607951473, −0.76935020799118005436249496265, −0.34343656218144123503072271666, −0.22121468390301595157909422831, −0.05743720043484823956650769982, 0.05743720043484823956650769982, 0.22121468390301595157909422831, 0.34343656218144123503072271666, 0.76935020799118005436249496265, 0.882724409831705366553607951473, 0.936199096923585792646386629966, 1.03592345230120291762210919898, 1.27567932158929819208969679113, 1.29191832746272161862674408278, 1.52579825895698779457666885764, 1.52751249273918298430174049297, 1.86637093752516451751141309969, 1.87856591472453537333759649943, 2.05120183344966994532669281005, 2.12488819413898098326702039720, 2.25977559065271480881644531148, 2.26776952378465629430712287428, 2.47445277991148694205461537381, 2.51824631695324832158276186123, 2.67260383093966421076349675398, 2.72064048450740367774580464262, 2.77269385001125963418677357324, 3.18939853358993259507778788679, 3.19069248639252601088189347410, 3.22179582374106985687948057846

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

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