Properties

Label 16-12e24-1.1-c2e8-0-7
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $2.41562\times 10^{13}$
Root an. cond. $6.86182$
Motivic weight $2$
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  + 3·5-s + 3·7-s + 18·11-s − 5·13-s − 6·17-s + 81·23-s + 43·25-s + 69·29-s + 45·31-s + 9·35-s + 20·37-s − 54·41-s − 207·47-s − 73·49-s − 252·53-s + 54·55-s − 306·59-s − 7·61-s − 15·65-s − 12·67-s + 74·73-s + 54·77-s + 33·79-s + 549·83-s − 18·85-s + 168·89-s − 15·91-s + ⋯
L(s)  = 1  + 3/5·5-s + 3/7·7-s + 1.63·11-s − 0.384·13-s − 0.352·17-s + 3.52·23-s + 1.71·25-s + 2.37·29-s + 1.45·31-s + 9/35·35-s + 0.540·37-s − 1.31·41-s − 4.40·47-s − 1.48·49-s − 4.75·53-s + 0.981·55-s − 5.18·59-s − 0.114·61-s − 0.230·65-s − 0.179·67-s + 1.01·73-s + 0.701·77-s + 0.417·79-s + 6.61·83-s − 0.211·85-s + 1.88·89-s − 0.164·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(2.41562\times 10^{13}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.045061611\)
\(L(\frac12)\) \(\approx\) \(2.045061611\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 - 3 T - 34 T^{2} + 339 T^{3} + 211 T^{4} - 10296 T^{5} + 50708 T^{6} + 156576 T^{7} - 1757804 T^{8} + 156576 p^{2} T^{9} + 50708 p^{4} T^{10} - 10296 p^{6} T^{11} + 211 p^{8} T^{12} + 339 p^{10} T^{13} - 34 p^{12} T^{14} - 3 p^{14} T^{15} + p^{16} T^{16} \)
7 \( 1 - 3 T + 82 T^{2} - 237 T^{3} + 2419 T^{4} - 21600 T^{5} + 116980 T^{6} - 2348256 T^{7} + 7641628 T^{8} - 2348256 p^{2} T^{9} + 116980 p^{4} T^{10} - 21600 p^{6} T^{11} + 2419 p^{8} T^{12} - 237 p^{10} T^{13} + 82 p^{12} T^{14} - 3 p^{14} T^{15} + p^{16} T^{16} \)
11 \( 1 - 18 T + 448 T^{2} - 6120 T^{3} + 93463 T^{4} - 1253556 T^{5} + 16013428 T^{6} - 201806226 T^{7} + 2222076136 T^{8} - 201806226 p^{2} T^{9} + 16013428 p^{4} T^{10} - 1253556 p^{6} T^{11} + 93463 p^{8} T^{12} - 6120 p^{10} T^{13} + 448 p^{12} T^{14} - 18 p^{14} T^{15} + p^{16} T^{16} \)
13 \( 1 + 5 T - 360 T^{2} + 1561 T^{3} + 77501 T^{4} - 597312 T^{5} - 5007086 T^{6} + 5241266 p T^{7} + 60552144 T^{8} + 5241266 p^{3} T^{9} - 5007086 p^{4} T^{10} - 597312 p^{6} T^{11} + 77501 p^{8} T^{12} + 1561 p^{10} T^{13} - 360 p^{12} T^{14} + 5 p^{14} T^{15} + p^{16} T^{16} \)
17 \( ( 1 + 3 T + 334 T^{2} + 693 T^{3} + 110178 T^{4} + 693 p^{2} T^{5} + 334 p^{4} T^{6} + 3 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
19 \( 1 - 1157 T^{2} + 639226 T^{4} - 232952123 T^{6} + 79410905002 T^{8} - 232952123 p^{4} T^{10} + 639226 p^{8} T^{12} - 1157 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 - 81 T + 4870 T^{2} - 217323 T^{3} + 8414383 T^{4} - 278460504 T^{5} + 8325324592 T^{6} - 221238967356 T^{7} + 5380294547716 T^{8} - 221238967356 p^{2} T^{9} + 8325324592 p^{4} T^{10} - 278460504 p^{6} T^{11} + 8414383 p^{8} T^{12} - 217323 p^{10} T^{13} + 4870 p^{12} T^{14} - 81 p^{14} T^{15} + p^{16} T^{16} \)
29 \( 1 - 69 T + 1976 T^{2} - 82905 T^{3} + 2728573 T^{4} - 24459264 T^{5} + 249830930 T^{6} + 10215187926 T^{7} - 1249058812400 T^{8} + 10215187926 p^{2} T^{9} + 249830930 p^{4} T^{10} - 24459264 p^{6} T^{11} + 2728573 p^{8} T^{12} - 82905 p^{10} T^{13} + 1976 p^{12} T^{14} - 69 p^{14} T^{15} + p^{16} T^{16} \)
31 \( 1 - 45 T + 3520 T^{2} - 128025 T^{3} + 6239425 T^{4} - 221987952 T^{5} + 7979134318 T^{6} - 268603930890 T^{7} + 8040333650992 T^{8} - 268603930890 p^{2} T^{9} + 7979134318 p^{4} T^{10} - 221987952 p^{6} T^{11} + 6239425 p^{8} T^{12} - 128025 p^{10} T^{13} + 3520 p^{12} T^{14} - 45 p^{14} T^{15} + p^{16} T^{16} \)
37 \( ( 1 - 10 T + 1720 T^{2} + 76250 T^{3} + 347470 T^{4} + 76250 p^{2} T^{5} + 1720 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( 1 + 54 T - 1108 T^{2} + 54648 T^{3} + 6254527 T^{4} - 135074304 T^{5} - 346116544 T^{6} + 221393044530 T^{7} - 6391866656792 T^{8} + 221393044530 p^{2} T^{9} - 346116544 p^{4} T^{10} - 135074304 p^{6} T^{11} + 6254527 p^{8} T^{12} + 54648 p^{10} T^{13} - 1108 p^{12} T^{14} + 54 p^{14} T^{15} + p^{16} T^{16} \)
43 \( 1 + 4210 T^{2} + 9163465 T^{4} - 282271716 T^{5} + 12018514642 T^{6} - 1023475278060 T^{7} + 9810745543012 T^{8} - 1023475278060 p^{2} T^{9} + 12018514642 p^{4} T^{10} - 282271716 p^{6} T^{11} + 9163465 p^{8} T^{12} + 4210 p^{12} T^{14} + p^{16} T^{16} \)
47 \( 1 + 207 T + 24406 T^{2} + 2095461 T^{3} + 142092151 T^{4} + 7940274480 T^{5} + 383592144688 T^{6} + 17253891231516 T^{7} + 786328390118740 T^{8} + 17253891231516 p^{2} T^{9} + 383592144688 p^{4} T^{10} + 7940274480 p^{6} T^{11} + 142092151 p^{8} T^{12} + 2095461 p^{10} T^{13} + 24406 p^{12} T^{14} + 207 p^{14} T^{15} + p^{16} T^{16} \)
53 \( ( 1 + 126 T + 12208 T^{2} + 787698 T^{3} + 46295070 T^{4} + 787698 p^{2} T^{5} + 12208 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( 1 + 306 T + 52768 T^{2} + 6596136 T^{3} + 655249567 T^{4} + 54840345300 T^{5} + 4031461665844 T^{6} + 267864455425266 T^{7} + 16404403358884120 T^{8} + 267864455425266 p^{2} T^{9} + 4031461665844 p^{4} T^{10} + 54840345300 p^{6} T^{11} + 655249567 p^{8} T^{12} + 6596136 p^{10} T^{13} + 52768 p^{12} T^{14} + 306 p^{14} T^{15} + p^{16} T^{16} \)
61 \( 1 + 7 T - 8478 T^{2} + 347909 T^{3} + 38508983 T^{4} - 2257992504 T^{5} - 39583245248 T^{6} + 5257966541956 T^{7} - 30952228546836 T^{8} + 5257966541956 p^{2} T^{9} - 39583245248 p^{4} T^{10} - 2257992504 p^{6} T^{11} + 38508983 p^{8} T^{12} + 347909 p^{10} T^{13} - 8478 p^{12} T^{14} + 7 p^{14} T^{15} + p^{16} T^{16} \)
67 \( 1 + 12 T + 11866 T^{2} + 141816 T^{3} + 66010801 T^{4} + 721613340 T^{5} + 400272803818 T^{6} + 3347530167216 T^{7} + 2268475959596500 T^{8} + 3347530167216 p^{2} T^{9} + 400272803818 p^{4} T^{10} + 721613340 p^{6} T^{11} + 66010801 p^{8} T^{12} + 141816 p^{10} T^{13} + 11866 p^{12} T^{14} + 12 p^{14} T^{15} + p^{16} T^{16} \)
71 \( 1 - 14120 T^{2} + 150082588 T^{4} - 1080637518872 T^{6} + 6319440918414790 T^{8} - 1080637518872 p^{4} T^{10} + 150082588 p^{8} T^{12} - 14120 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 - 37 T + 20314 T^{2} - 565891 T^{3} + 160126666 T^{4} - 565891 p^{2} T^{5} + 20314 p^{4} T^{6} - 37 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 - 33 T + 13924 T^{2} - 447513 T^{3} + 83250565 T^{4} - 4145960160 T^{5} + 461050019242 T^{6} - 35083235957766 T^{7} + 3234248047096936 T^{8} - 35083235957766 p^{2} T^{9} + 461050019242 p^{4} T^{10} - 4145960160 p^{6} T^{11} + 83250565 p^{8} T^{12} - 447513 p^{10} T^{13} + 13924 p^{12} T^{14} - 33 p^{14} T^{15} + p^{16} T^{16} \)
83 \( 1 - 549 T + 157876 T^{2} - 31517541 T^{3} + 4848813277 T^{4} - 610140548160 T^{5} + 65668976887426 T^{6} - 6260783050033902 T^{7} + 542374558465652968 T^{8} - 6260783050033902 p^{2} T^{9} + 65668976887426 p^{4} T^{10} - 610140548160 p^{6} T^{11} + 4848813277 p^{8} T^{12} - 31517541 p^{10} T^{13} + 157876 p^{12} T^{14} - 549 p^{14} T^{15} + p^{16} T^{16} \)
89 \( ( 1 - 84 T + 30700 T^{2} - 1886940 T^{3} + 359702982 T^{4} - 1886940 p^{2} T^{5} + 30700 p^{4} T^{6} - 84 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( 1 + 10 T - 22548 T^{2} + 828296 T^{3} + 272325791 T^{4} - 14518511616 T^{5} - 1388516520704 T^{6} + 83318070893038 T^{7} + 5416086623373288 T^{8} + 83318070893038 p^{2} T^{9} - 1388516520704 p^{4} T^{10} - 14518511616 p^{6} T^{11} + 272325791 p^{8} T^{12} + 828296 p^{10} T^{13} - 22548 p^{12} T^{14} + 10 p^{14} T^{15} + p^{16} T^{16} \)
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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

−3.52199931009206927810265480907, −3.46483619865865914607273146993, −3.42019807464164893340684103275, −3.34230857948160390884356900099, −3.22554197126219033946388426611, −3.18631868244848556563572557528, −2.93743834435003252649324578005, −2.85578083957115533896955190482, −2.76423993544227592175591324495, −2.67448695060971532711790239060, −2.36786377089504614721729966092, −2.34165468615677444931789327780, −2.07098046591698700813434348143, −2.03156032408632842507639308337, −1.69224713044837702362012688491, −1.62696637143667994111095450337, −1.51916620324430889346853008534, −1.25207095466556157835306417846, −1.22647464713998315601285775920, −1.21278326629309510341090974873, −1.10483712779617980452660631171, −0.71547188191975716822892170898, −0.45046221253912567995158382555, −0.43924493086832258358779703722, −0.06636453876717912340577243844, 0.06636453876717912340577243844, 0.43924493086832258358779703722, 0.45046221253912567995158382555, 0.71547188191975716822892170898, 1.10483712779617980452660631171, 1.21278326629309510341090974873, 1.22647464713998315601285775920, 1.25207095466556157835306417846, 1.51916620324430889346853008534, 1.62696637143667994111095450337, 1.69224713044837702362012688491, 2.03156032408632842507639308337, 2.07098046591698700813434348143, 2.34165468615677444931789327780, 2.36786377089504614721729966092, 2.67448695060971532711790239060, 2.76423993544227592175591324495, 2.85578083957115533896955190482, 2.93743834435003252649324578005, 3.18631868244848556563572557528, 3.22554197126219033946388426611, 3.34230857948160390884356900099, 3.42019807464164893340684103275, 3.46483619865865914607273146993, 3.52199931009206927810265480907

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

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