Properties

Label 16-65e8-1.1-c1e8-0-2
Degree $16$
Conductor $3.186\times 10^{14}$
Sign $1$
Analytic cond. $0.00526651$
Root an. cond. $0.720435$
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  − 6·3-s + 4·4-s − 2·5-s + 18·9-s + 6·11-s − 24·12-s + 14·13-s + 12·15-s + 2·16-s + 16·17-s − 14·19-s − 8·20-s − 14·23-s − 4·25-s − 38·27-s + 2·31-s − 36·33-s + 72·36-s − 44·37-s − 84·39-s + 16·41-s − 6·43-s + 24·44-s − 36·45-s + 16·47-s − 12·48-s − 16·49-s + ⋯
L(s)  = 1  − 3.46·3-s + 2·4-s − 0.894·5-s + 6·9-s + 1.80·11-s − 6.92·12-s + 3.88·13-s + 3.09·15-s + 1/2·16-s + 3.88·17-s − 3.21·19-s − 1.78·20-s − 2.91·23-s − 4/5·25-s − 7.31·27-s + 0.359·31-s − 6.26·33-s + 12·36-s − 7.23·37-s − 13.4·39-s + 2.49·41-s − 0.914·43-s + 3.61·44-s − 5.36·45-s + 2.33·47-s − 1.73·48-s − 2.28·49-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(5^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(0.00526651\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2232416048\)
\(L(\frac12)\) \(\approx\) \(0.2232416048\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 2 T + 8 T^{2} + 6 T^{3} + 38 T^{4} + 6 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13 \( 1 - 14 T + 100 T^{2} - 538 T^{3} + 2254 T^{4} - 538 p T^{5} + 100 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
good2 \( 1 - p^{2} T^{2} + 7 p T^{4} - 5 p^{3} T^{6} + 81 T^{8} - 5 p^{5} T^{10} + 7 p^{5} T^{12} - p^{8} T^{14} + p^{8} T^{16} \)
3 \( 1 + 2 p T + 2 p^{2} T^{2} + 38 T^{3} + 68 T^{4} + 122 T^{5} + 230 T^{6} + 418 T^{7} + 730 T^{8} + 418 p T^{9} + 230 p^{2} T^{10} + 122 p^{3} T^{11} + 68 p^{4} T^{12} + 38 p^{5} T^{13} + 2 p^{8} T^{14} + 2 p^{8} T^{15} + p^{8} T^{16} \)
7 \( ( 1 + 8 T^{2} + 4 T^{3} + 66 T^{4} + 4 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( 1 - 6 T + 18 T^{2} - 74 T^{3} + 480 T^{4} - 1902 T^{5} + 5510 T^{6} - 22402 T^{7} + 90754 T^{8} - 22402 p T^{9} + 5510 p^{2} T^{10} - 1902 p^{3} T^{11} + 480 p^{4} T^{12} - 74 p^{5} T^{13} + 18 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 16 T + 128 T^{2} - 808 T^{3} + 4588 T^{4} - 1320 p T^{5} + 98208 T^{6} - 412960 T^{7} + 1709478 T^{8} - 412960 p T^{9} + 98208 p^{2} T^{10} - 1320 p^{4} T^{11} + 4588 p^{4} T^{12} - 808 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 + 14 T + 98 T^{2} + 662 T^{3} + 4880 T^{4} + 28162 T^{5} + 135150 T^{6} + 689194 T^{7} + 3344898 T^{8} + 689194 p T^{9} + 135150 p^{2} T^{10} + 28162 p^{3} T^{11} + 4880 p^{4} T^{12} + 662 p^{5} T^{13} + 98 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 + 14 T + 98 T^{2} + 610 T^{3} + 3540 T^{4} + 16574 T^{5} + 71166 T^{6} + 303274 T^{7} + 1330874 T^{8} + 303274 p T^{9} + 71166 p^{2} T^{10} + 16574 p^{3} T^{11} + 3540 p^{4} T^{12} + 610 p^{5} T^{13} + 98 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 188 T^{2} + 16572 T^{4} - 885220 T^{6} + 31234358 T^{8} - 885220 p^{2} T^{10} + 16572 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} \)
31 \( 1 - 2 T + 2 T^{2} - 30 T^{3} + 1184 T^{4} - 94 p T^{5} + 3910 T^{6} - 89742 T^{7} + 2057506 T^{8} - 89742 p T^{9} + 3910 p^{2} T^{10} - 94 p^{4} T^{11} + 1184 p^{4} T^{12} - 30 p^{5} T^{13} + 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
37 \( ( 1 + 22 T + 320 T^{2} + 2994 T^{3} + 21522 T^{4} + 2994 p T^{5} + 320 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 - 16 T + 128 T^{2} - 744 T^{3} + 5852 T^{4} - 58120 T^{5} + 457632 T^{6} - 2817600 T^{7} + 17080198 T^{8} - 2817600 p T^{9} + 457632 p^{2} T^{10} - 58120 p^{3} T^{11} + 5852 p^{4} T^{12} - 744 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 + 6 T + 18 T^{2} - 2 p T^{3} + 596 T^{4} + 17470 T^{5} + 97790 T^{6} + 211786 T^{7} - 2781606 T^{8} + 211786 p T^{9} + 97790 p^{2} T^{10} + 17470 p^{3} T^{11} + 596 p^{4} T^{12} - 2 p^{6} T^{13} + 18 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
47 \( ( 1 - 8 T + 140 T^{2} - 764 T^{3} + 8578 T^{4} - 764 p T^{5} + 140 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( 1 + 24 T + 288 T^{2} + 2936 T^{3} + 30556 T^{4} + 284840 T^{5} + 2346080 T^{6} + 19062856 T^{7} + 147591846 T^{8} + 19062856 p T^{9} + 2346080 p^{2} T^{10} + 284840 p^{3} T^{11} + 30556 p^{4} T^{12} + 2936 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + 22 T + 242 T^{2} + 1870 T^{3} + 8144 T^{4} + 22682 T^{5} + 276606 T^{6} + 4745202 T^{7} + 49894594 T^{8} + 4745202 p T^{9} + 276606 p^{2} T^{10} + 22682 p^{3} T^{11} + 8144 p^{4} T^{12} + 1870 p^{5} T^{13} + 242 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \)
61 \( ( 1 - 10 T + 124 T^{2} - 318 T^{3} + 4058 T^{4} - 318 p T^{5} + 124 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 436 T^{2} + 88524 T^{4} - 10894412 T^{6} + 887067734 T^{8} - 10894412 p^{2} T^{10} + 88524 p^{4} T^{12} - 436 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 + 10 T + 50 T^{2} + 818 T^{3} + 56 p T^{4} - 33914 T^{5} - 203378 T^{6} - 3779058 T^{7} - 67309230 T^{8} - 3779058 p T^{9} - 203378 p^{2} T^{10} - 33914 p^{3} T^{11} + 56 p^{5} T^{12} + 818 p^{5} T^{13} + 50 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 468 T^{2} + 103244 T^{4} - 13839756 T^{6} + 1228100854 T^{8} - 13839756 p^{2} T^{10} + 103244 p^{4} T^{12} - 468 p^{6} T^{14} + p^{8} T^{16} \)
79 \( 1 - 340 T^{2} + 65892 T^{4} - 8434412 T^{6} + 781259126 T^{8} - 8434412 p^{2} T^{10} + 65892 p^{4} T^{12} - 340 p^{6} T^{14} + p^{8} T^{16} \)
83 \( ( 1 - 24 T + 412 T^{2} - 4596 T^{3} + 47242 T^{4} - 4596 p T^{5} + 412 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 - 28 T + 392 T^{2} - 4020 T^{3} + 31916 T^{4} - 289180 T^{5} + 3666168 T^{6} - 49450068 T^{7} + 555566950 T^{8} - 49450068 p T^{9} + 3666168 p^{2} T^{10} - 289180 p^{3} T^{11} + 31916 p^{4} T^{12} - 4020 p^{5} T^{13} + 392 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - 484 T^{2} + 123060 T^{4} - 20185820 T^{6} + 2320434518 T^{8} - 20185820 p^{2} T^{10} + 123060 p^{4} T^{12} - 484 p^{6} T^{14} + p^{8} T^{16} \)
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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

−6.95738544924012154504552569643, −6.95064798054050482535908870810, −6.58916540207676816303569402893, −6.42631704887152813800309218563, −6.38173171427100255901988622293, −6.31482594562809561032123816117, −6.29517315320610496333082381339, −6.00339715772345686883820357953, −5.84079827862216822915270236847, −5.74518930177726512045151808561, −5.51807968751059160155402231676, −5.16352461134965396108962758127, −5.05012988023387437334995194351, −4.93592658142669140209693027403, −4.41656751633675708093808512305, −4.15968694170517662130271696138, −4.06033931094218668455165127899, −3.58694210607745382157200566185, −3.56539913698672882849639035954, −3.55001204275113155740720305607, −3.45819930895912012042979199078, −2.47263790372287207439622527512, −1.76097132527405707291291299262, −1.68593253710102619942642065110, −1.59490065267106985890189734947, 1.59490065267106985890189734947, 1.68593253710102619942642065110, 1.76097132527405707291291299262, 2.47263790372287207439622527512, 3.45819930895912012042979199078, 3.55001204275113155740720305607, 3.56539913698672882849639035954, 3.58694210607745382157200566185, 4.06033931094218668455165127899, 4.15968694170517662130271696138, 4.41656751633675708093808512305, 4.93592658142669140209693027403, 5.05012988023387437334995194351, 5.16352461134965396108962758127, 5.51807968751059160155402231676, 5.74518930177726512045151808561, 5.84079827862216822915270236847, 6.00339715772345686883820357953, 6.29517315320610496333082381339, 6.31482594562809561032123816117, 6.38173171427100255901988622293, 6.42631704887152813800309218563, 6.58916540207676816303569402893, 6.95064798054050482535908870810, 6.95738544924012154504552569643

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

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