Properties

Degree $16$
Conductor $2.732\times 10^{25}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 2·7-s − 2·11-s − 8·13-s + 4·17-s − 6·19-s + 2·23-s + 20·25-s + 16·29-s − 6·31-s + 8·35-s − 16·41-s − 20·47-s − 49-s − 10·53-s + 8·55-s + 22·59-s + 2·61-s + 32·65-s + 2·67-s + 44·71-s − 10·73-s + 4·77-s + 8·79-s − 40·83-s − 16·85-s − 16·89-s + ⋯
L(s)  = 1  − 1.78·5-s − 0.755·7-s − 0.603·11-s − 2.21·13-s + 0.970·17-s − 1.37·19-s + 0.417·23-s + 4·25-s + 2.97·29-s − 1.07·31-s + 1.35·35-s − 2.49·41-s − 2.91·47-s − 1/7·49-s − 1.37·53-s + 1.07·55-s + 2.86·59-s + 0.256·61-s + 3.96·65-s + 0.244·67-s + 5.22·71-s − 1.17·73-s + 0.455·77-s + 0.900·79-s − 4.39·83-s − 1.73·85-s − 1.69·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{24} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1512} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.04628883190\)
\(L(\frac12)\) \(\approx\) \(0.04628883190\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + 2 T + 5 T^{2} + 18 T^{3} + 8 T^{4} + 18 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
good5 \( ( 1 + 2 T - 4 T^{2} - 4 T^{3} + 19 T^{4} - 4 p T^{5} - 4 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( 1 + 2 T - 14 T^{2} - 48 T^{3} - 58 T^{4} + 26 T^{5} - 168 T^{6} + 2902 T^{7} + 26203 T^{8} + 2902 p T^{9} - 168 p^{2} T^{10} + 26 p^{3} T^{11} - 58 p^{4} T^{12} - 48 p^{5} T^{13} - 14 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
13 \( ( 1 + 4 T + 17 T^{2} - 48 T^{3} - 160 T^{4} - 48 p T^{5} + 17 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 2 T - 4 T^{2} + 52 T^{3} - 293 T^{4} + 52 p T^{5} - 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + 877 T^{4} - 936 T^{5} + 9706 T^{6} + 54060 T^{7} - 159614 T^{8} + 54060 p T^{9} + 9706 p^{2} T^{10} - 936 p^{3} T^{11} + 877 p^{4} T^{12} + 6 p^{6} T^{13} - 5 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 2 T - 10 T^{2} + 72 T^{3} - 922 T^{4} + 2222 T^{5} + 760 T^{6} - 57846 T^{7} + 692635 T^{8} - 57846 p T^{9} + 760 p^{2} T^{10} + 2222 p^{3} T^{11} - 922 p^{4} T^{12} + 72 p^{5} T^{13} - 10 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
29 \( ( 1 - 8 T + 36 T^{2} - 24 T^{3} - 182 T^{4} - 24 p T^{5} + 36 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 + 6 T - 38 T^{2} - 348 T^{3} + 517 T^{4} + 10584 T^{5} + 34246 T^{6} - 188550 T^{7} - 2230412 T^{8} - 188550 p T^{9} + 34246 p^{2} T^{10} + 10584 p^{3} T^{11} + 517 p^{4} T^{12} - 348 p^{5} T^{13} - 38 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 - 61 T^{2} - 648 T^{3} + 2017 T^{4} + 31752 T^{5} + 168050 T^{6} - 910440 T^{7} - 7948706 T^{8} - 910440 p T^{9} + 168050 p^{2} T^{10} + 31752 p^{3} T^{11} + 2017 p^{4} T^{12} - 648 p^{5} T^{13} - 61 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 8 T + 60 T^{2} + 456 T^{3} + 2554 T^{4} + 456 p T^{5} + 60 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 + 14 T^{2} - 72 T^{3} - 1881 T^{4} - 72 p T^{5} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( 1 + 20 T + 196 T^{2} + 1464 T^{3} + 7286 T^{4} + 1508 T^{5} - 284064 T^{6} - 3306668 T^{7} - 27412733 T^{8} - 3306668 p T^{9} - 284064 p^{2} T^{10} + 1508 p^{3} T^{11} + 7286 p^{4} T^{12} + 1464 p^{5} T^{13} + 196 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 10 T - 34 T^{2} - 720 T^{3} - 538 T^{4} + 28730 T^{5} + 169816 T^{6} - 937470 T^{7} - 17639549 T^{8} - 937470 p T^{9} + 169816 p^{2} T^{10} + 28730 p^{3} T^{11} - 538 p^{4} T^{12} - 720 p^{5} T^{13} - 34 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 22 T + 106 T^{2} - 48 T^{3} + 14486 T^{4} - 158758 T^{5} + 199560 T^{6} - 3891698 T^{7} + 83505403 T^{8} - 3891698 p T^{9} + 199560 p^{2} T^{10} - 158758 p^{3} T^{11} + 14486 p^{4} T^{12} - 48 p^{5} T^{13} + 106 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 2 T - 90 T^{2} + 116 T^{3} + 4577 T^{4} - 7740 T^{5} + 332338 T^{6} - 11534 T^{7} - 30676068 T^{8} - 11534 p T^{9} + 332338 p^{2} T^{10} - 7740 p^{3} T^{11} + 4577 p^{4} T^{12} + 116 p^{5} T^{13} - 90 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 2 T - 145 T^{2} - 310 T^{3} + 10229 T^{4} + 49232 T^{5} - 447054 T^{6} - 1987140 T^{7} + 26208538 T^{8} - 1987140 p T^{9} - 447054 p^{2} T^{10} + 49232 p^{3} T^{11} + 10229 p^{4} T^{12} - 310 p^{5} T^{13} - 145 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
71 \( ( 1 - 22 T + 402 T^{2} - 4638 T^{3} + 45946 T^{4} - 4638 p T^{5} + 402 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 + 10 T - 141 T^{2} - 1714 T^{3} + 12677 T^{4} + 151788 T^{5} - 657782 T^{6} - 4715864 T^{7} + 43753554 T^{8} - 4715864 p T^{9} - 657782 p^{2} T^{10} + 151788 p^{3} T^{11} + 12677 p^{4} T^{12} - 1714 p^{5} T^{13} - 141 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 8 T - 201 T^{2} + 920 T^{3} + 28853 T^{4} - 55488 T^{5} - 3083690 T^{6} + 2064928 T^{7} + 256704030 T^{8} + 2064928 p T^{9} - 3083690 p^{2} T^{10} - 55488 p^{3} T^{11} + 28853 p^{4} T^{12} + 920 p^{5} T^{13} - 201 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
83 \( ( 1 + 20 T + 360 T^{2} + 3684 T^{3} + 40318 T^{4} + 3684 p T^{5} + 360 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 + 16 T - 92 T^{2} - 1248 T^{3} + 22646 T^{4} + 83536 T^{5} - 3013152 T^{6} - 6008080 T^{7} + 235094707 T^{8} - 6008080 p T^{9} - 3013152 p^{2} T^{10} + 83536 p^{3} T^{11} + 22646 p^{4} T^{12} - 1248 p^{5} T^{13} - 92 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
97 \( ( 1 + 230 T^{2} + 72 T^{3} + 26415 T^{4} + 72 p T^{5} + 230 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−4.04602859476724519563555549999, −3.89142713102203108391099714886, −3.87321869016964869947796854362, −3.58167610752748162065782769150, −3.57111728208576798152175456248, −3.37914285349999702716660866080, −3.14872905933703799236455747580, −3.04992072483740888816267154942, −2.98806446747878580260446857904, −2.86767121388427997037417498206, −2.80341582084159481311282125308, −2.62117604846453655452189502923, −2.47839516799113917776833262370, −2.43001701036444273426795257735, −2.18490873170643708916146804000, −2.00674587904678495629981535704, −1.95071449623415159785052764685, −1.52394781064912974943033983237, −1.30514332682071178552361819466, −1.27169706354273796400999819832, −1.17362554084108333378873687218, −0.955432690791957019430356584792, −0.56900562117707439395768722492, −0.15341856744821742687036161406, −0.084616886721849132447205533697, 0.084616886721849132447205533697, 0.15341856744821742687036161406, 0.56900562117707439395768722492, 0.955432690791957019430356584792, 1.17362554084108333378873687218, 1.27169706354273796400999819832, 1.30514332682071178552361819466, 1.52394781064912974943033983237, 1.95071449623415159785052764685, 2.00674587904678495629981535704, 2.18490873170643708916146804000, 2.43001701036444273426795257735, 2.47839516799113917776833262370, 2.62117604846453655452189502923, 2.80341582084159481311282125308, 2.86767121388427997037417498206, 2.98806446747878580260446857904, 3.04992072483740888816267154942, 3.14872905933703799236455747580, 3.37914285349999702716660866080, 3.57111728208576798152175456248, 3.58167610752748162065782769150, 3.87321869016964869947796854362, 3.89142713102203108391099714886, 4.04602859476724519563555549999

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

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