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Properties

Label 16-91e16-1.1-c1e8-0-1
Degree $16$
Conductor $2.211\times 10^{31}$
Sign $1$
Analytic cond. $3.65493\times 10^{14}$
Root an. cond. $8.13167$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $8$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 5·4-s + 2·9-s + 20·12-s + 9·16-s − 8·17-s − 12·23-s − 20·25-s + 12·27-s − 8·29-s − 10·36-s − 8·43-s − 36·48-s + 32·51-s + 20·53-s − 12·61-s − 10·64-s + 40·68-s + 48·69-s + 80·75-s − 20·79-s − 17·81-s + 32·87-s + 60·92-s + 100·100-s + 8·101-s − 32·103-s + ⋯
L(s)  = 1  − 2.30·3-s − 5/2·4-s + 2/3·9-s + 5.77·12-s + 9/4·16-s − 1.94·17-s − 2.50·23-s − 4·25-s + 2.30·27-s − 1.48·29-s − 5/3·36-s − 1.21·43-s − 5.19·48-s + 4.48·51-s + 2.74·53-s − 1.53·61-s − 5/4·64-s + 4.85·68-s + 5.77·69-s + 9.23·75-s − 2.25·79-s − 1.88·81-s + 3.43·87-s + 6.25·92-s + 10·100-s + 0.796·101-s − 3.15·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(7^{16} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(3.65493\times 10^{14}\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(8\)
Selberg data: \((16,\ 7^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + 5 T^{2} + p^{4} T^{4} + 45 T^{6} + 103 T^{8} + 45 p^{2} T^{10} + p^{8} T^{12} + 5 p^{6} T^{14} + p^{8} T^{16} \)
3 \( ( 1 + 2 T + 5 T^{2} + 8 T^{3} + 14 T^{4} + 8 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
5 \( 1 + 4 p T^{2} + 203 T^{4} + 56 p^{2} T^{6} + 7648 T^{8} + 56 p^{4} T^{10} + 203 p^{4} T^{12} + 4 p^{7} T^{14} + p^{8} T^{16} \)
11 \( 1 + 36 T^{2} + 552 T^{4} + 6040 T^{6} + 65873 T^{8} + 6040 p^{2} T^{10} + 552 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + 4 T + 48 T^{2} + 152 T^{3} + 1177 T^{4} + 152 p T^{5} + 48 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 + 108 T^{2} + 5632 T^{4} + 184552 T^{6} + 4170841 T^{8} + 184552 p^{2} T^{10} + 5632 p^{4} T^{12} + 108 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 + 6 T + 97 T^{2} + 404 T^{3} + 3398 T^{4} + 404 p T^{5} + 97 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 4 T + 53 T^{2} + 140 T^{3} + 2016 T^{4} + 140 p T^{5} + 53 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 + 168 T^{2} + 14119 T^{4} + 754888 T^{6} + 27862780 T^{8} + 754888 p^{2} T^{10} + 14119 p^{4} T^{12} + 168 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 + 176 T^{2} + 16039 T^{4} + 966152 T^{6} + 41670748 T^{8} + 966152 p^{2} T^{10} + 16039 p^{4} T^{12} + 176 p^{6} T^{14} + p^{8} T^{16} \)
41 \( 1 + 196 T^{2} + 20328 T^{4} + 1383052 T^{6} + 66725070 T^{8} + 1383052 p^{2} T^{10} + 20328 p^{4} T^{12} + 196 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 + 4 T + 106 T^{2} + 672 T^{3} + 5314 T^{4} + 672 p T^{5} + 106 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 + 180 T^{2} + 18792 T^{4} + 1371592 T^{6} + 73763777 T^{8} + 1371592 p^{2} T^{10} + 18792 p^{4} T^{12} + 180 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 - 10 T + 4 p T^{2} - 1460 T^{3} + 16941 T^{4} - 1460 p T^{5} + 4 p^{3} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 + 284 T^{2} + 41392 T^{4} + 3995400 T^{6} + 275818393 T^{8} + 3995400 p^{2} T^{10} + 41392 p^{4} T^{12} + 284 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 + 6 T + 220 T^{2} + 1004 T^{3} + 19621 T^{4} + 1004 p T^{5} + 220 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 252 T^{2} + 33176 T^{4} + 3237176 T^{6} + 247467313 T^{8} + 3237176 p^{2} T^{10} + 33176 p^{4} T^{12} + 252 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 + 276 T^{2} + 36585 T^{4} + 3341788 T^{6} + 252824924 T^{8} + 3341788 p^{2} T^{10} + 36585 p^{4} T^{12} + 276 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 + 324 T^{2} + 55307 T^{4} + 6362312 T^{6} + 537626848 T^{8} + 6362312 p^{2} T^{10} + 55307 p^{4} T^{12} + 324 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + 10 T + 325 T^{2} + 2310 T^{3} + 38860 T^{4} + 2310 p T^{5} + 325 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 368 T^{2} + 74176 T^{4} + 9998736 T^{6} + 969195550 T^{8} + 9998736 p^{2} T^{10} + 74176 p^{4} T^{12} + 368 p^{6} T^{14} + p^{8} T^{16} \)
89 \( 1 + 272 T^{2} + 49727 T^{4} + 6652904 T^{6} + 662082940 T^{8} + 6652904 p^{2} T^{10} + 49727 p^{4} T^{12} + 272 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 + 672 T^{2} + 205664 T^{4} + 37489952 T^{6} + 4452439678 T^{8} + 37489952 p^{2} T^{10} + 205664 p^{4} T^{12} + 672 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.63657282694831383454113344092, −3.61251896242794407979096330499, −3.36513141073420227044169073972, −3.24093440863252038807681725215, −3.20302053033433480360608456622, −3.06575732070285242249069495097, −3.02177012672839222540862299562, −2.97014467910985241613886011348, −2.61281183600614638552906187075, −2.47039312055297003544782886301, −2.44231730706419414208011681800, −2.34894022211159490656854624017, −2.33860090423227444633063435027, −2.08775715148758506069072048486, −1.98206467383696375068401962370, −1.97390992123833741769853430374, −1.90990942800528622667509103629, −1.71824944918448313662234595346, −1.61038822606495055853458712143, −1.33582963065993274254226516207, −1.15334739931615011682030791655, −1.09278405736923749504238365935, −1.02121165064616773052687666808, −0.833638563461437303505595916894, −0.73306930527171164673927438902, 0, 0, 0, 0, 0, 0, 0, 0, 0.73306930527171164673927438902, 0.833638563461437303505595916894, 1.02121165064616773052687666808, 1.09278405736923749504238365935, 1.15334739931615011682030791655, 1.33582963065993274254226516207, 1.61038822606495055853458712143, 1.71824944918448313662234595346, 1.90990942800528622667509103629, 1.97390992123833741769853430374, 1.98206467383696375068401962370, 2.08775715148758506069072048486, 2.33860090423227444633063435027, 2.34894022211159490656854624017, 2.44231730706419414208011681800, 2.47039312055297003544782886301, 2.61281183600614638552906187075, 2.97014467910985241613886011348, 3.02177012672839222540862299562, 3.06575732070285242249069495097, 3.20302053033433480360608456622, 3.24093440863252038807681725215, 3.36513141073420227044169073972, 3.61251896242794407979096330499, 3.63657282694831383454113344092

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

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