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Properties

Label 16-5e32-1.1-c1e8-0-12
Degree $16$
Conductor $2.328\times 10^{22}$
Sign $1$
Analytic cond. $384819.$
Root an. cond. $2.23397$
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  + 5·4-s + 10·9-s + 16·11-s + 9·16-s − 10·19-s − 20·29-s + 16·31-s + 50·36-s + 26·41-s + 80·44-s + 35·49-s − 30·59-s + 6·61-s + 10·64-s + 46·71-s − 50·76-s − 10·79-s + 39·81-s − 30·89-s + 160·99-s + 26·101-s + 10·109-s − 100·116-s + 56·121-s + 80·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 5/2·4-s + 10/3·9-s + 4.82·11-s + 9/4·16-s − 2.29·19-s − 3.71·29-s + 2.87·31-s + 25/3·36-s + 4.06·41-s + 12.0·44-s + 5·49-s − 3.90·59-s + 0.768·61-s + 5/4·64-s + 5.45·71-s − 5.73·76-s − 1.12·79-s + 13/3·81-s − 3.17·89-s + 16.0·99-s + 2.58·101-s + 0.957·109-s − 9.28·116-s + 5.09·121-s + 7.18·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{32}\)
Sign: $1$
Analytic conductor: \(384819.\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(39.28802176\)
\(L(\frac12)\) \(\approx\) \(39.28802176\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 - 5 T^{2} + p^{4} T^{4} - 45 T^{6} + 101 T^{8} - 45 p^{2} T^{10} + p^{8} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} \)
3 \( 1 - 10 T^{2} + 61 T^{4} - 10 p^{3} T^{6} + 916 T^{8} - 10 p^{5} T^{10} + 61 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 - 5 p T^{2} + 611 T^{4} - 7045 T^{6} + 57976 T^{8} - 7045 p^{2} T^{10} + 611 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} \)
11 \( ( 1 - 2 T + p T^{2} )^{8} \)
13 \( 1 - 90 T^{2} + 3671 T^{4} - 89140 T^{6} + 1415181 T^{8} - 89140 p^{2} T^{10} + 3671 p^{4} T^{12} - 90 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 95 T^{2} + 4351 T^{4} - 125745 T^{6} + 2528816 T^{8} - 125745 p^{2} T^{10} + 4351 p^{4} T^{12} - 95 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 5 T + 71 T^{2} + 255 T^{3} + 1956 T^{4} + 255 p T^{5} + 71 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 150 T^{2} + 10421 T^{4} - 438710 T^{6} + 12245916 T^{8} - 438710 p^{2} T^{10} + 10421 p^{4} T^{12} - 150 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 10 T + 111 T^{2} + 20 p T^{3} + 4061 T^{4} + 20 p^{2} T^{5} + 111 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 8 T + 83 T^{2} - 416 T^{3} + 3180 T^{4} - 416 p T^{5} + 83 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 5 p T^{2} + 17246 T^{4} - 28220 p T^{6} + 45156381 T^{8} - 28220 p^{3} T^{10} + 17246 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} \)
41 \( ( 1 - 13 T + 183 T^{2} - 1451 T^{3} + 11760 T^{4} - 1451 p T^{5} + 183 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 5 p T^{2} + 22911 T^{4} - 1578205 T^{6} + 78597176 T^{8} - 1578205 p^{2} T^{10} + 22911 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} \)
47 \( 1 - 5 p T^{2} + 26751 T^{4} - 1970705 T^{6} + 106163336 T^{8} - 1970705 p^{2} T^{10} + 26751 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} \)
53 \( 1 - 185 T^{2} + 23006 T^{4} - 1859700 T^{6} + 115939701 T^{8} - 1859700 p^{2} T^{10} + 23006 p^{4} T^{12} - 185 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 15 T + 241 T^{2} + 2025 T^{3} + 19456 T^{4} + 2025 p T^{5} + 241 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 3 T + 98 T^{2} - 786 T^{3} + 4855 T^{4} - 786 p T^{5} + 98 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 360 T^{2} + 62716 T^{4} - 6991000 T^{6} + 550222886 T^{8} - 6991000 p^{2} T^{10} + 62716 p^{4} T^{12} - 360 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 23 T + 383 T^{2} - 4101 T^{3} + 39380 T^{4} - 4101 p T^{5} + 383 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 505 T^{2} + 114686 T^{4} - 15492260 T^{6} + 1375657261 T^{8} - 15492260 p^{2} T^{10} + 114686 p^{4} T^{12} - 505 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + 5 T + 121 T^{2} + 15 p T^{3} + 12416 T^{4} + 15 p^{2} T^{5} + 121 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 290 T^{2} + 41701 T^{4} - 4874870 T^{6} + 471491716 T^{8} - 4874870 p^{2} T^{10} + 41701 p^{4} T^{12} - 290 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 15 T + 321 T^{2} + 3475 T^{3} + 42476 T^{4} + 3475 p T^{5} + 321 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 110 T^{2} + 22811 T^{4} - 2099540 T^{6} + 303117741 T^{8} - 2099540 p^{2} T^{10} + 22811 p^{4} T^{12} - 110 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.48527994715935859344201098111, −4.40239656489755064252275269882, −4.28505296981207434430536691210, −4.06316884268715968887955230691, −4.00523220122841880762150490796, −3.93229807316909968410896507126, −3.84111926204662469073651999540, −3.83458301320954607802188359650, −3.76192568949082443731332114637, −3.54962190012892801322421520682, −2.96711445854223092087639059557, −2.94865898953985717761374478965, −2.92866501857315221728458968662, −2.66344085466365775473730107319, −2.39653268886373042952949246236, −2.31701566371914172604145336869, −2.01870587524153959122999305401, −1.97260617274964958684803338868, −1.89940276215716482269348278087, −1.62350423096045406912601506138, −1.54459962503501652282022184823, −1.30303342505675785986481450924, −0.917299398637956091300305279931, −0.891605398386285778221091692255, −0.77216903789538782553680636537, 0.77216903789538782553680636537, 0.891605398386285778221091692255, 0.917299398637956091300305279931, 1.30303342505675785986481450924, 1.54459962503501652282022184823, 1.62350423096045406912601506138, 1.89940276215716482269348278087, 1.97260617274964958684803338868, 2.01870587524153959122999305401, 2.31701566371914172604145336869, 2.39653268886373042952949246236, 2.66344085466365775473730107319, 2.92866501857315221728458968662, 2.94865898953985717761374478965, 2.96711445854223092087639059557, 3.54962190012892801322421520682, 3.76192568949082443731332114637, 3.83458301320954607802188359650, 3.84111926204662469073651999540, 3.93229807316909968410896507126, 4.00523220122841880762150490796, 4.06316884268715968887955230691, 4.28505296981207434430536691210, 4.40239656489755064252275269882, 4.48527994715935859344201098111

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

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