Properties

Label 16-42e16-1.1-c2e8-0-0
Degree $16$
Conductor $9.375\times 10^{25}$
Sign $1$
Analytic cond. $2.84884\times 10^{13}$
Root an. cond. $6.93293$
Motivic weight $2$
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  + 16·13-s − 40·19-s − 68·25-s + 40·31-s − 32·37-s − 32·43-s − 272·61-s − 168·67-s + 144·73-s − 232·79-s − 192·97-s − 328·103-s − 240·109-s − 368·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 760·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 1.23·13-s − 2.10·19-s − 2.71·25-s + 1.29·31-s − 0.864·37-s − 0.744·43-s − 4.45·61-s − 2.50·67-s + 1.97·73-s − 2.93·79-s − 1.97·97-s − 3.18·103-s − 2.20·109-s − 3.04·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4.49·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(2.84884\times 10^{13}\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.007738254147\)
\(L(\frac12)\) \(\approx\) \(0.007738254147\)
\(L(2)\) 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 \( 1 + 68 T^{2} + 466 p T^{4} + 70992 T^{6} + 1981811 T^{8} + 70992 p^{4} T^{10} + 466 p^{9} T^{12} + 68 p^{12} T^{14} + p^{16} T^{16} \)
11 \( 1 + 368 T^{2} + 72734 T^{4} + 12294144 T^{6} + 1732934435 T^{8} + 12294144 p^{4} T^{10} + 72734 p^{8} T^{12} + 368 p^{12} T^{14} + p^{16} T^{16} \)
13 \( ( 1 - 4 T + 230 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
17 \( 1 + 452 T^{2} - 11014 T^{4} + 21820752 T^{6} + 22322409299 T^{8} + 21820752 p^{4} T^{10} - 11014 p^{8} T^{12} + 452 p^{12} T^{14} + p^{16} T^{16} \)
19 \( ( 1 + 20 T - 170 T^{2} - 160 p T^{3} + 139 p^{2} T^{4} - 160 p^{3} T^{5} - 170 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
23 \( 1 + 1808 T^{2} + 1913918 T^{4} + 1437837312 T^{6} + 842805507011 T^{8} + 1437837312 p^{4} T^{10} + 1913918 p^{8} T^{12} + 1808 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 + 112 T^{2} - 475102 T^{4} + 112 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 - 20 T - 1594 T^{2} - 1440 T^{3} + 2668115 T^{4} - 1440 p^{2} T^{5} - 1594 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( ( 1 + 16 T - 1538 T^{2} - 15104 T^{3} + 993811 T^{4} - 15104 p^{2} T^{5} - 1538 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( ( 1 - 6436 T^{2} + 15997974 T^{4} - 6436 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 + 8 T + 2706 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
47 \( 1 + 2644 T^{2} + 709162 T^{4} - 9195271472 T^{6} - 18465676379501 T^{8} - 9195271472 p^{4} T^{10} + 709162 p^{8} T^{12} + 2644 p^{12} T^{14} + p^{16} T^{16} \)
53 \( ( 1 + 3440 T^{2} + 3943119 T^{4} + 3440 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
59 \( 1 + 6836 T^{2} + 19594250 T^{4} + 19837552464 T^{6} + 23311981839539 T^{8} + 19837552464 p^{4} T^{10} + 19594250 p^{8} T^{12} + 6836 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 + 136 T + 6458 T^{2} + 625056 T^{3} + 62243987 T^{4} + 625056 p^{2} T^{5} + 6458 p^{4} T^{6} + 136 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( ( 1 + 84 T + 1802 T^{2} - 312816 T^{3} - 24220989 T^{4} - 312816 p^{2} T^{5} + 1802 p^{4} T^{6} + 84 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 4496 T^{2} + 27200834 T^{4} - 4496 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 - 72 T - 6742 T^{2} - 91296 T^{3} + 86205699 T^{4} - 91296 p^{2} T^{5} - 6742 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( ( 1 + 116 T + 4778 T^{2} - 441264 T^{3} - 47621293 T^{4} - 441264 p^{2} T^{5} + 4778 p^{4} T^{6} + 116 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( ( 1 - 8228 T^{2} + 33565286 T^{4} - 8228 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( 1 - 17852 T^{2} + 125955514 T^{4} - 1200616765616 T^{6} + 13215928444091155 T^{8} - 1200616765616 p^{4} T^{10} + 125955514 p^{8} T^{12} - 17852 p^{12} T^{14} + p^{16} T^{16} \)
97 \( ( 1 + 48 T + 11302 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
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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

−3.69802606827863488797441943710, −3.67517097963199837578949898298, −3.52802626670471235810033564477, −3.38558412352894232207873786085, −3.28154603822496780679877376659, −3.06034666499668755255563224965, −2.83449135857306516574452887261, −2.71038960915827705592273429316, −2.70401793885347107763848699332, −2.56551428214542547093369505818, −2.42843282234642482022492467373, −2.41354126081058226096149712489, −2.20077333846730210329843650925, −1.81542590051898427621862449651, −1.81268721949698941319463305847, −1.59199838621251135683914554809, −1.53477721793857067283798187901, −1.33517757274874330546604158311, −1.31562163329042823133068513089, −1.29854636209180201016643701182, −0.974231474497562281604677898644, −0.60113403828317595424369085794, −0.33945423412825129150540254760, −0.10589148357740278200294727633, −0.02353551905975386068164230444, 0.02353551905975386068164230444, 0.10589148357740278200294727633, 0.33945423412825129150540254760, 0.60113403828317595424369085794, 0.974231474497562281604677898644, 1.29854636209180201016643701182, 1.31562163329042823133068513089, 1.33517757274874330546604158311, 1.53477721793857067283798187901, 1.59199838621251135683914554809, 1.81268721949698941319463305847, 1.81542590051898427621862449651, 2.20077333846730210329843650925, 2.41354126081058226096149712489, 2.42843282234642482022492467373, 2.56551428214542547093369505818, 2.70401793885347107763848699332, 2.71038960915827705592273429316, 2.83449135857306516574452887261, 3.06034666499668755255563224965, 3.28154603822496780679877376659, 3.38558412352894232207873786085, 3.52802626670471235810033564477, 3.67517097963199837578949898298, 3.69802606827863488797441943710

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

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