Properties

Label 16-378e8-1.1-c3e8-0-3
Degree $16$
Conductor $4.168\times 10^{20}$
Sign $1$
Analytic cond. $6.12157\times 10^{10}$
Root an. cond. $4.72257$
Motivic weight $3$
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  − 8·2-s + 24·4-s + 2·5-s + 6·7-s − 16·10-s − 32·11-s − 4·13-s − 48·14-s − 240·16-s + 58·17-s + 70·19-s + 48·20-s + 256·22-s − 86·23-s + 174·25-s + 32·26-s + 144·28-s + 212·29-s − 64·31-s + 768·32-s − 464·34-s + 12·35-s − 146·37-s − 560·38-s + 780·41-s + 880·43-s − 768·44-s + ⋯
L(s)  = 1  − 2.82·2-s + 3·4-s + 0.178·5-s + 0.323·7-s − 0.505·10-s − 0.877·11-s − 0.0853·13-s − 0.916·14-s − 3.75·16-s + 0.827·17-s + 0.845·19-s + 0.536·20-s + 2.48·22-s − 0.779·23-s + 1.39·25-s + 0.241·26-s + 0.971·28-s + 1.35·29-s − 0.370·31-s + 4.24·32-s − 2.34·34-s + 0.0579·35-s − 0.648·37-s − 2.39·38-s + 2.97·41-s + 3.12·43-s − 2.63·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{24} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(6.12157\times 10^{10}\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.1078740926\)
\(L(\frac12)\) \(\approx\) \(0.1078740926\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p T + p^{2} T^{2} )^{4} \)
3 \( 1 \)
7 \( 1 - 6 T - p T^{2} - 354 p T^{3} - 1140 p^{2} T^{4} - 354 p^{4} T^{5} - p^{7} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
good5 \( 1 - 2 T - 34 p T^{2} + 912 T^{3} - 6922 T^{4} - 44618 T^{5} - 576312 T^{6} - 915938 p T^{7} + 540349219 T^{8} - 915938 p^{4} T^{9} - 576312 p^{6} T^{10} - 44618 p^{9} T^{11} - 6922 p^{12} T^{12} + 912 p^{15} T^{13} - 34 p^{19} T^{14} - 2 p^{21} T^{15} + p^{24} T^{16} \)
11 \( 1 + 32 T - 3524 T^{2} - 77568 T^{3} + 8319434 T^{4} + 92921024 T^{5} - 16193852112 T^{6} - 36132028832 T^{7} + 25840028173315 T^{8} - 36132028832 p^{3} T^{9} - 16193852112 p^{6} T^{10} + 92921024 p^{9} T^{11} + 8319434 p^{12} T^{12} - 77568 p^{15} T^{13} - 3524 p^{18} T^{14} + 32 p^{21} T^{15} + p^{24} T^{16} \)
13 \( ( 1 + 2 T + 3905 T^{2} - 134190 T^{3} + 6991376 T^{4} - 134190 p^{3} T^{5} + 3905 p^{6} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
17 \( 1 - 58 T - 8702 T^{2} + 1253208 T^{3} + 17600774 T^{4} - 488456042 p T^{5} + 396908983752 T^{6} + 22332126339646 T^{7} - 2893130337334877 T^{8} + 22332126339646 p^{3} T^{9} + 396908983752 p^{6} T^{10} - 488456042 p^{10} T^{11} + 17600774 p^{12} T^{12} + 1253208 p^{15} T^{13} - 8702 p^{18} T^{14} - 58 p^{21} T^{15} + p^{24} T^{16} \)
19 \( 1 - 70 T - 12273 T^{2} + 1205422 T^{3} + 45848873 T^{4} - 308262288 p T^{5} - 343174472426 T^{6} + 6059827390508 T^{7} + 4294435442748282 T^{8} + 6059827390508 p^{3} T^{9} - 343174472426 p^{6} T^{10} - 308262288 p^{10} T^{11} + 45848873 p^{12} T^{12} + 1205422 p^{15} T^{13} - 12273 p^{18} T^{14} - 70 p^{21} T^{15} + p^{24} T^{16} \)
23 \( 1 + 86 T - 39890 T^{2} - 82632 p T^{3} + 1124840390 T^{4} + 30401287838 T^{5} - 20627892302808 T^{6} - 126588682789742 T^{7} + 295654711270877179 T^{8} - 126588682789742 p^{3} T^{9} - 20627892302808 p^{6} T^{10} + 30401287838 p^{9} T^{11} + 1124840390 p^{12} T^{12} - 82632 p^{16} T^{13} - 39890 p^{18} T^{14} + 86 p^{21} T^{15} + p^{24} T^{16} \)
29 \( ( 1 - 106 T + 58386 T^{2} - 2107854 T^{3} + 1482347746 T^{4} - 2107854 p^{3} T^{5} + 58386 p^{6} T^{6} - 106 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
31 \( 1 + 64 T - 75334 T^{2} - 4471888 T^{3} + 3046133861 T^{4} + 143301706760 T^{5} - 80185819225710 T^{6} - 2278462920223944 T^{7} + 1894674291144946204 T^{8} - 2278462920223944 p^{3} T^{9} - 80185819225710 p^{6} T^{10} + 143301706760 p^{9} T^{11} + 3046133861 p^{12} T^{12} - 4471888 p^{15} T^{13} - 75334 p^{18} T^{14} + 64 p^{21} T^{15} + p^{24} T^{16} \)
37 \( 1 + 146 T - 41037 T^{2} + 24382222 T^{3} + 4523622941 T^{4} - 1003339821252 T^{5} + 404910943728514 T^{6} + 86923322843591240 T^{7} - 13743755171151605190 T^{8} + 86923322843591240 p^{3} T^{9} + 404910943728514 p^{6} T^{10} - 1003339821252 p^{9} T^{11} + 4523622941 p^{12} T^{12} + 24382222 p^{15} T^{13} - 41037 p^{18} T^{14} + 146 p^{21} T^{15} + p^{24} T^{16} \)
41 \( ( 1 - 390 T + 214610 T^{2} - 60532506 T^{3} + 22214903298 T^{4} - 60532506 p^{3} T^{5} + 214610 p^{6} T^{6} - 390 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
43 \( ( 1 - 440 T + 352918 T^{2} - 101685224 T^{3} + 43426524751 T^{4} - 101685224 p^{3} T^{5} + 352918 p^{6} T^{6} - 440 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
47 \( 1 - 306 T - 73958 T^{2} - 59983056 T^{3} + 24769315990 T^{4} + 4026015675846 T^{5} + 2990954943459448 T^{6} - 21838254530911650 p T^{7} - 85885331168320805 p^{2} T^{8} - 21838254530911650 p^{4} T^{9} + 2990954943459448 p^{6} T^{10} + 4026015675846 p^{9} T^{11} + 24769315990 p^{12} T^{12} - 59983056 p^{15} T^{13} - 73958 p^{18} T^{14} - 306 p^{21} T^{15} + p^{24} T^{16} \)
53 \( 1 - 90 T - 111650 T^{2} + 55596816 T^{3} - 21791681306 T^{4} - 2948884766586 T^{5} + 1995977814148024 T^{6} - 452517461869649058 T^{7} + \)\(17\!\cdots\!27\)\( T^{8} - 452517461869649058 p^{3} T^{9} + 1995977814148024 p^{6} T^{10} - 2948884766586 p^{9} T^{11} - 21791681306 p^{12} T^{12} + 55596816 p^{15} T^{13} - 111650 p^{18} T^{14} - 90 p^{21} T^{15} + p^{24} T^{16} \)
59 \( 1 + 148 T - 724172 T^{2} - 46183224 T^{3} + 322992258026 T^{4} + 8773729679068 T^{5} - 98974299460087056 T^{6} - 669594626352126556 T^{7} + \)\(23\!\cdots\!15\)\( T^{8} - 669594626352126556 p^{3} T^{9} - 98974299460087056 p^{6} T^{10} + 8773729679068 p^{9} T^{11} + 322992258026 p^{12} T^{12} - 46183224 p^{15} T^{13} - 724172 p^{18} T^{14} + 148 p^{21} T^{15} + p^{24} T^{16} \)
61 \( 1 + 364 T - 627802 T^{2} - 224697736 T^{3} + 3996544229 p T^{4} + 1157855553464 p T^{5} - 64264165920142506 T^{6} - 7095447872603585772 T^{7} + \)\(15\!\cdots\!52\)\( T^{8} - 7095447872603585772 p^{3} T^{9} - 64264165920142506 p^{6} T^{10} + 1157855553464 p^{10} T^{11} + 3996544229 p^{13} T^{12} - 224697736 p^{15} T^{13} - 627802 p^{18} T^{14} + 364 p^{21} T^{15} + p^{24} T^{16} \)
67 \( 1 + 954 T - 528629 T^{2} - 389300634 T^{3} + 544584174805 T^{4} + 225418821262920 T^{5} - 201962106024878270 T^{6} - 8847830048260684788 T^{7} + \)\(89\!\cdots\!70\)\( T^{8} - 8847830048260684788 p^{3} T^{9} - 201962106024878270 p^{6} T^{10} + 225418821262920 p^{9} T^{11} + 544584174805 p^{12} T^{12} - 389300634 p^{15} T^{13} - 528629 p^{18} T^{14} + 954 p^{21} T^{15} + p^{24} T^{16} \)
71 \( ( 1 - 680 T + 1000056 T^{2} - 625468776 T^{3} + 454321868590 T^{4} - 625468776 p^{3} T^{5} + 1000056 p^{6} T^{6} - 680 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
73 \( 1 + 54 T - 356153 T^{2} - 344801358 T^{3} + 80712341209 T^{4} + 132077540633940 T^{5} + 124229498711894494 T^{6} - 37768506737759197488 T^{7} - \)\(53\!\cdots\!66\)\( T^{8} - 37768506737759197488 p^{3} T^{9} + 124229498711894494 p^{6} T^{10} + 132077540633940 p^{9} T^{11} + 80712341209 p^{12} T^{12} - 344801358 p^{15} T^{13} - 356153 p^{18} T^{14} + 54 p^{21} T^{15} + p^{24} T^{16} \)
79 \( 1 + 226 T - 487693 T^{2} - 344699266 T^{3} - 260271124975 T^{4} + 15628583326928 T^{5} + 22127295683141214 T^{6} + 48887116558883288988 T^{7} + \)\(10\!\cdots\!02\)\( T^{8} + 48887116558883288988 p^{3} T^{9} + 22127295683141214 p^{6} T^{10} + 15628583326928 p^{9} T^{11} - 260271124975 p^{12} T^{12} - 344699266 p^{15} T^{13} - 487693 p^{18} T^{14} + 226 p^{21} T^{15} + p^{24} T^{16} \)
83 \( ( 1 - 1568 T + 1587240 T^{2} - 1167350640 T^{3} + 850483913182 T^{4} - 1167350640 p^{3} T^{5} + 1587240 p^{6} T^{6} - 1568 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
89 \( 1 - 1458 T + 542326 T^{2} - 83239488 T^{3} + 189980218918 T^{4} - 519794357813514 T^{5} + 567863209418477128 T^{6} + 52698436128215759790 T^{7} - \)\(39\!\cdots\!25\)\( T^{8} + 52698436128215759790 p^{3} T^{9} + 567863209418477128 p^{6} T^{10} - 519794357813514 p^{9} T^{11} + 189980218918 p^{12} T^{12} - 83239488 p^{15} T^{13} + 542326 p^{18} T^{14} - 1458 p^{21} T^{15} + p^{24} T^{16} \)
97 \( ( 1 + 2172 T + 2538830 T^{2} + 2100215688 T^{3} + 1807738972095 T^{4} + 2100215688 p^{3} T^{5} + 2538830 p^{6} T^{6} + 2172 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
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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

−4.55654270820787179689170492389, −4.40290535774035179804260385174, −4.22885670231816296852392775433, −4.22412896915155233071957611361, −3.97539787840570871065495780072, −3.67575673650430195678563500715, −3.64790515804719430762761649167, −3.45934114131511748812381403713, −3.42070518334339030680791431052, −2.98059259254825355828263148618, −2.80694422639756788280229813070, −2.72092022479224643587743597031, −2.50138204065484523785146700870, −2.37163233684309076971032413052, −2.28718808642070899351304118311, −2.07583499518602712202093137218, −1.85373158349401491232864318264, −1.48871774508796350006167096584, −1.12972654615193328749387550844, −1.04732456912772378976196074720, −1.01795088789252218560507228784, −0.983860103837046622524081920467, −0.78540110117526812469970809782, −0.15306492401934372753366707601, −0.12544291981785664860003456181, 0.12544291981785664860003456181, 0.15306492401934372753366707601, 0.78540110117526812469970809782, 0.983860103837046622524081920467, 1.01795088789252218560507228784, 1.04732456912772378976196074720, 1.12972654615193328749387550844, 1.48871774508796350006167096584, 1.85373158349401491232864318264, 2.07583499518602712202093137218, 2.28718808642070899351304118311, 2.37163233684309076971032413052, 2.50138204065484523785146700870, 2.72092022479224643587743597031, 2.80694422639756788280229813070, 2.98059259254825355828263148618, 3.42070518334339030680791431052, 3.45934114131511748812381403713, 3.64790515804719430762761649167, 3.67575673650430195678563500715, 3.97539787840570871065495780072, 4.22412896915155233071957611361, 4.22885670231816296852392775433, 4.40290535774035179804260385174, 4.55654270820787179689170492389

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

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