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Properties

Label 16-845e8-1.1-c1e8-0-13
Degree $16$
Conductor $2.599\times 10^{23}$
Sign $1$
Analytic cond. $4.29606\times 10^{6}$
Root an. cond. $2.59756$
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 + 2·3-s + 5·4-s + 8·5-s + 4·6-s + 10·7-s + 6·8-s + 6·9-s + 16·10-s + 10·12-s + 20·14-s + 16·15-s + 9·16-s + 2·17-s + 12·18-s + 16·19-s + 40·20-s + 20·21-s + 10·23-s + 12·24-s + 36·25-s + 12·27-s + 50·28-s − 8·29-s + 32·30-s − 16·31-s + 6·32-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 5/2·4-s + 3.57·5-s + 1.63·6-s + 3.77·7-s + 2.12·8-s + 2·9-s + 5.05·10-s + 2.88·12-s + 5.34·14-s + 4.13·15-s + 9/4·16-s + 0.485·17-s + 2.82·18-s + 3.67·19-s + 8.94·20-s + 4.36·21-s + 2.08·23-s + 2.44·24-s + 36/5·25-s + 2.30·27-s + 9.44·28-s − 1.48·29-s + 5.84·30-s − 2.87·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{8} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(4.29606\times 10^{6}\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{8} \cdot 13^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(377.8503098\)
\(L(\frac12)\) \(\approx\) \(377.8503098\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 - T )^{8} \)
13 \( 1 \)
good2 \( ( 1 - p^{2} T + 5 T^{2} + p T^{3} - 11 T^{4} + p^{2} T^{5} + 5 p^{2} T^{6} - p^{5} T^{7} + p^{4} T^{8} )( 1 + p T + p T^{2} + p T^{3} + T^{4} + p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} ) \)
3 \( 1 - 2 T - 2 T^{2} + 4 T^{3} + T^{4} + 4 T^{5} + 10 T^{6} - 26 T^{7} - 20 T^{8} - 26 p T^{9} + 10 p^{2} T^{10} + 4 p^{3} T^{11} + p^{4} T^{12} + 4 p^{5} T^{13} - 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
7 \( 1 - 10 T + 6 p T^{2} - 116 T^{3} + 341 T^{4} - 1020 T^{5} + 2494 T^{6} - 6562 T^{7} + 18804 T^{8} - 6562 p T^{9} + 2494 p^{2} T^{10} - 1020 p^{3} T^{11} + 341 p^{4} T^{12} - 116 p^{5} T^{13} + 6 p^{7} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 - 14 T^{2} + 97 T^{4} + 182 p T^{6} - 236 p^{2} T^{8} + 182 p^{3} T^{10} + 97 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 - 16 T + 128 T^{2} - 688 T^{3} + 3022 T^{4} - 688 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )( 1 + 14 T + 50 T^{2} - 220 T^{3} - 2129 T^{4} - 220 p T^{5} + 50 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} ) \)
19 \( ( 1 - 8 T + 13 T^{2} - 104 T^{3} + 1024 T^{4} - 104 p T^{5} + 13 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 10 T + 2 T^{2} + 108 T^{3} + 1013 T^{4} - 2804 T^{5} - 34298 T^{6} + 4266 p T^{7} + 8492 p T^{8} + 4266 p^{2} T^{9} - 34298 p^{2} T^{10} - 2804 p^{3} T^{11} + 1013 p^{4} T^{12} + 108 p^{5} T^{13} + 2 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 + 8 T - 34 T^{2} - 528 T^{3} + 353 T^{4} + 19264 T^{5} + 50686 T^{6} - 248280 T^{7} - 2118188 T^{8} - 248280 p T^{9} + 50686 p^{2} T^{10} + 19264 p^{3} T^{11} + 353 p^{4} T^{12} - 528 p^{5} T^{13} - 34 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
31 \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
37 \( 1 + 2 T - 90 T^{2} - 332 T^{3} + 4097 T^{4} + 17184 T^{5} - 90854 T^{6} - 352942 T^{7} + 1733340 T^{8} - 352942 p T^{9} - 90854 p^{2} T^{10} + 17184 p^{3} T^{11} + 4097 p^{4} T^{12} - 332 p^{5} T^{13} - 90 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
41 \( ( 1 - 4 T - 67 T^{2} - 4 T^{3} + 4552 T^{4} - 4 p T^{5} - 67 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 2 T - 150 T^{2} + 188 T^{3} + 13661 T^{4} - 10620 T^{5} - 862418 T^{6} + 185374 T^{7} + 42096180 T^{8} + 185374 p T^{9} - 862418 p^{2} T^{10} - 10620 p^{3} T^{11} + 13661 p^{4} T^{12} + 188 p^{5} T^{13} - 150 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
47 \( ( 1 + 8 T + 116 T^{2} + 392 T^{3} + 5158 T^{4} + 392 p T^{5} + 116 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + 12 T + 248 T^{2} + 36 p T^{3} + 20622 T^{4} + 36 p^{2} T^{5} + 248 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 12 T - 122 T^{2} + 1080 T^{3} + 20701 T^{4} - 108048 T^{5} - 1594418 T^{6} + 1301340 T^{7} + 128025820 T^{8} + 1301340 p T^{9} - 1594418 p^{2} T^{10} - 108048 p^{3} T^{11} + 20701 p^{4} T^{12} + 1080 p^{5} T^{13} - 122 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 28 T + 282 T^{2} + 1880 T^{3} + 26477 T^{4} + 344016 T^{5} + 2759698 T^{6} + 21027916 T^{7} + 175399068 T^{8} + 21027916 p T^{9} + 2759698 p^{2} T^{10} + 344016 p^{3} T^{11} + 26477 p^{4} T^{12} + 1880 p^{5} T^{13} + 282 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 30 T + 302 T^{2} - 2724 T^{3} + 55441 T^{4} - 623940 T^{5} + 3752474 T^{6} - 42317478 T^{7} + 502476364 T^{8} - 42317478 p T^{9} + 3752474 p^{2} T^{10} - 623940 p^{3} T^{11} + 55441 p^{4} T^{12} - 2724 p^{5} T^{13} + 302 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 4 T - 58 T^{2} + 552 T^{3} - 7123 T^{4} + 25744 T^{5} - 1394 T^{6} - 3091260 T^{7} + 64548268 T^{8} - 3091260 p T^{9} - 1394 p^{2} T^{10} + 25744 p^{3} T^{11} - 7123 p^{4} T^{12} + 552 p^{5} T^{13} - 58 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
73 \( ( 1 - 8 T + 208 T^{2} - 920 T^{3} + 17998 T^{4} - 920 p T^{5} + 208 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 8 T + 184 T^{2} + 1256 T^{3} + 21022 T^{4} + 1256 p T^{5} + 184 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( ( 1 - 12 T + 308 T^{2} - 2700 T^{3} + 37158 T^{4} - 2700 p T^{5} + 308 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 + 12 T + 22 T^{2} - 648 T^{3} - 12515 T^{4} - 130632 T^{5} + 123358 T^{6} + 12675588 T^{7} + 140030092 T^{8} + 12675588 p T^{9} + 123358 p^{2} T^{10} - 130632 p^{3} T^{11} - 12515 p^{4} T^{12} - 648 p^{5} T^{13} + 22 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - 2 T - 294 T^{2} + 1316 T^{3} + 48341 T^{4} - 223416 T^{5} - 5151962 T^{6} + 10980406 T^{7} + 480743076 T^{8} + 10980406 p T^{9} - 5151962 p^{2} T^{10} - 223416 p^{3} T^{11} + 48341 p^{4} T^{12} + 1316 p^{5} T^{13} - 294 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.43707968915906208272078157396, −4.14199261643933297261626175206, −4.08934130050012725805770056991, −3.92047663743441391393373328231, −3.88537009694859947603213335699, −3.75788023557850010252427638885, −3.74816151523346979431383218311, −3.33817253373553143532737467845, −3.15038477548417246592115623155, −3.01123355536095690793032785882, −2.90554181022745163427847394903, −2.86345210748665858378528249489, −2.69525534628405342916649396025, −2.68147540004336838645447792804, −2.48858660882722406099603404601, −1.96389422138464514973938544498, −1.91711729101855349514306535483, −1.87377430814152765989430284289, −1.80825382508330438466410831280, −1.67002722809890081136681214572, −1.63215417421461518845402725940, −1.30388454774815225141967179358, −1.17730105056341723671484210050, −0.978379207436018532948088861221, −0.851816672661626482220210886722, 0.851816672661626482220210886722, 0.978379207436018532948088861221, 1.17730105056341723671484210050, 1.30388454774815225141967179358, 1.63215417421461518845402725940, 1.67002722809890081136681214572, 1.80825382508330438466410831280, 1.87377430814152765989430284289, 1.91711729101855349514306535483, 1.96389422138464514973938544498, 2.48858660882722406099603404601, 2.68147540004336838645447792804, 2.69525534628405342916649396025, 2.86345210748665858378528249489, 2.90554181022745163427847394903, 3.01123355536095690793032785882, 3.15038477548417246592115623155, 3.33817253373553143532737467845, 3.74816151523346979431383218311, 3.75788023557850010252427638885, 3.88537009694859947603213335699, 3.92047663743441391393373328231, 4.08934130050012725805770056991, 4.14199261643933297261626175206, 4.43707968915906208272078157396

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

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