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Properties

Label 16-605e8-1.1-c1e8-0-12
Degree $16$
Conductor $1.795\times 10^{22}$
Sign $1$
Analytic cond. $296660.$
Root an. cond. $2.19794$
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  + 2·2-s + 3·4-s + 2·5-s − 4·7-s + 4·8-s − 2·9-s + 4·10-s − 8·13-s − 8·14-s + 4·16-s + 8·17-s − 4·18-s + 6·20-s + 25-s − 16·26-s − 12·28-s + 4·29-s + 18·32-s + 16·34-s − 8·35-s − 6·36-s + 4·37-s + 8·40-s + 12·41-s + 48·43-s − 4·45-s + 18·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.894·5-s − 1.51·7-s + 1.41·8-s − 2/3·9-s + 1.26·10-s − 2.21·13-s − 2.13·14-s + 16-s + 1.94·17-s − 0.942·18-s + 1.34·20-s + 1/5·25-s − 3.13·26-s − 2.26·28-s + 0.742·29-s + 3.18·32-s + 2.74·34-s − 1.35·35-s − 36-s + 0.657·37-s + 1.26·40-s + 1.87·41-s + 7.31·43-s − 0.596·45-s + 18/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{8} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(296660.\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(20.04392108\)
\(L(\frac12)\) \(\approx\) \(20.04392108\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
11 \( 1 \)
good2 \( 1 - p T + T^{2} + T^{4} - p^{4} T^{5} + 25 T^{6} - 3 p T^{7} - 3 T^{8} - 3 p^{2} T^{9} + 25 p^{2} T^{10} - p^{7} T^{11} + p^{4} T^{12} + p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \)
3 \( 1 + 2 T^{2} - 5 T^{4} - 28 T^{6} - 11 T^{8} - 28 p^{2} T^{10} - 5 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} \)
7 \( ( 1 + 2 T - 3 T^{2} - 20 T^{3} - 19 T^{4} - 20 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 + 8 T + 30 T^{2} + 72 T^{3} + 219 T^{4} + 264 T^{5} - 2916 T^{6} - 21696 T^{7} - 83979 T^{8} - 21696 p T^{9} - 2916 p^{2} T^{10} + 264 p^{3} T^{11} + 219 p^{4} T^{12} + 72 p^{5} T^{13} + 30 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 8 T + 22 T^{2} + 24 T^{3} - 317 T^{4} + 56 T^{5} + 3244 T^{6} + 7872 T^{7} - 115227 T^{8} + 7872 p T^{9} + 3244 p^{2} T^{10} + 56 p^{3} T^{11} - 317 p^{4} T^{12} + 24 p^{5} T^{13} + 22 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
19 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \)
29 \( 1 - 4 T - 14 T^{2} + 60 T^{3} - 197 T^{4} - 11540 T^{5} + 56884 T^{6} + 91056 T^{7} - 566427 T^{8} + 91056 p T^{9} + 56884 p^{2} T^{10} - 11540 p^{3} T^{11} - 197 p^{4} T^{12} + 60 p^{5} T^{13} - 14 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
31 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
37 \( 1 - 4 T - 30 T^{2} + 156 T^{3} - 21 T^{4} - 18132 T^{5} + 91476 T^{6} + 257712 T^{7} - 2526459 T^{8} + 257712 p T^{9} + 91476 p^{2} T^{10} - 18132 p^{3} T^{11} - 21 p^{4} T^{12} + 156 p^{5} T^{13} - 30 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
41 \( ( 1 - 6 T - 5 T^{2} + 276 T^{3} - 1451 T^{4} + 276 p T^{5} - 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 6 T + p T^{2} )^{8} \)
47 \( 1 - 86 T^{2} + 5187 T^{4} - 256108 T^{6} + 10567205 T^{8} - 256108 p^{2} T^{10} + 5187 p^{4} T^{12} - 86 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 12 T + 34 T^{2} - 276 T^{3} - 2229 T^{4} + 27516 T^{5} + 304340 T^{6} - 17040 T^{7} - 9920443 T^{8} - 17040 p T^{9} + 304340 p^{2} T^{10} + 27516 p^{3} T^{11} - 2229 p^{4} T^{12} - 276 p^{5} T^{13} + 34 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 8 T - 38 T^{2} + 648 T^{3} - 1013 T^{4} - 53320 T^{5} + 356308 T^{6} + 810624 T^{7} - 14135115 T^{8} + 810624 p T^{9} + 356308 p^{2} T^{10} - 53320 p^{3} T^{11} - 1013 p^{4} T^{12} + 648 p^{5} T^{13} - 38 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 4 T + 18 T^{2} - 324 T^{3} - 1413 T^{4} - 35220 T^{5} + 150132 T^{6} + 879408 T^{7} + 10634085 T^{8} + 879408 p T^{9} + 150132 p^{2} T^{10} - 35220 p^{3} T^{11} - 1413 p^{4} T^{12} - 324 p^{5} T^{13} + 18 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
67 \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
71 \( 1 - 14 T^{2} - 4845 T^{4} + 138404 T^{6} + 22485989 T^{8} + 138404 p^{2} T^{10} - 4845 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 + 8 T - 90 T^{2} - 1368 T^{3} + 2259 T^{4} + 22344 T^{5} - 488436 T^{6} + 1260864 T^{7} + 86316741 T^{8} + 1260864 p T^{9} - 488436 p^{2} T^{10} + 22344 p^{3} T^{11} + 2259 p^{4} T^{12} - 1368 p^{5} T^{13} - 90 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
79 \( ( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} + 568 p T^{5} - 63 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 6 T - 47 T^{2} - 780 T^{3} - 779 T^{4} - 780 p T^{5} - 47 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
97 \( 1 - 4 T - 150 T^{2} + 876 T^{3} + 13539 T^{4} - 83892 T^{5} - 840444 T^{6} + 3792432 T^{7} + 29505621 T^{8} + 3792432 p T^{9} - 840444 p^{2} T^{10} - 83892 p^{3} T^{11} + 13539 p^{4} T^{12} + 876 p^{5} T^{13} - 150 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.54144613373311084393936136669, −4.43464313202521559223413210483, −4.38779298882035372044621552175, −4.30285134970383774456270300170, −4.05154069083302940308054570042, −3.92349245121681516457866480384, −3.91326797233694192384758085417, −3.73734824564458086781724784553, −3.66203863264971816897911900887, −3.30850047495763793201893755807, −3.06936642421916171185677208465, −2.90424192660060011062777369118, −2.83376527832375832845454734170, −2.73684152176539910130351854145, −2.60428966022063993736069223832, −2.53853777692249772949488075715, −2.53473080909625189039432969968, −2.06582526720419860497490719606, −2.01023181642491046215374526281, −2.00044649559122557129723887904, −1.30301107886333873325601333491, −1.19155571365831732829406612797, −0.75639729838863468353415584226, −0.69997647689954684207234179931, −0.68690710746519975681180053912, 0.68690710746519975681180053912, 0.69997647689954684207234179931, 0.75639729838863468353415584226, 1.19155571365831732829406612797, 1.30301107886333873325601333491, 2.00044649559122557129723887904, 2.01023181642491046215374526281, 2.06582526720419860497490719606, 2.53473080909625189039432969968, 2.53853777692249772949488075715, 2.60428966022063993736069223832, 2.73684152176539910130351854145, 2.83376527832375832845454734170, 2.90424192660060011062777369118, 3.06936642421916171185677208465, 3.30850047495763793201893755807, 3.66203863264971816897911900887, 3.73734824564458086781724784553, 3.91326797233694192384758085417, 3.92349245121681516457866480384, 4.05154069083302940308054570042, 4.30285134970383774456270300170, 4.38779298882035372044621552175, 4.43464313202521559223413210483, 4.54144613373311084393936136669

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

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