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Properties

Label 16-5e32-1.1-c1e8-0-11
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

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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 − 3.33·9-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

\(L(1)\) \(\approx\) \(5.204944339\)
\(L(\frac12)\) \(\approx\) \(5.204944339\)
\(L(\frac{3}{2})\) 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

−4.78250557194392296936901799146, −4.44844740086080234971867279209, −4.18858206691464264526283857714, −4.17550371593200739780692679367, −4.09038173439617518390019754370, −3.98638222645220691928882734529, −3.81361346441448892059888956793, −3.78936711402077546722203334863, −3.67257849316276330073712996259, −3.20570169926995149903493540191, −3.13557225334101723193890946713, −3.08281597098021272038323946602, −2.96213855548454580222314276960, −2.93567704606131208737572349084, −2.71204763940202793146626237294, −2.42037034266320535485217120401, −2.12391743410827168387720789473, −2.05491715345655117625188202953, −1.91924333199322949496136727297, −1.32604388590584069230662337399, −1.27451764543905497032262795004, −0.913110224888650774189629819112, −0.71922159910025532786245851025, −0.70278862233976831519159038095, −0.66639282546174732916322857934, 0.66639282546174732916322857934, 0.70278862233976831519159038095, 0.71922159910025532786245851025, 0.913110224888650774189629819112, 1.27451764543905497032262795004, 1.32604388590584069230662337399, 1.91924333199322949496136727297, 2.05491715345655117625188202953, 2.12391743410827168387720789473, 2.42037034266320535485217120401, 2.71204763940202793146626237294, 2.93567704606131208737572349084, 2.96213855548454580222314276960, 3.08281597098021272038323946602, 3.13557225334101723193890946713, 3.20570169926995149903493540191, 3.67257849316276330073712996259, 3.78936711402077546722203334863, 3.81361346441448892059888956793, 3.98638222645220691928882734529, 4.09038173439617518390019754370, 4.17550371593200739780692679367, 4.18858206691464264526283857714, 4.44844740086080234971867279209, 4.78250557194392296936901799146

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

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