Properties

Degree $16$
Conductor $1.849\times 10^{17}$
Sign $1$
Motivic weight $2$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 3·5-s + 3·7-s + 3·9-s + 18·11-s + 5·13-s − 9·15-s + 6·17-s − 9·21-s − 81·23-s + 43·25-s + 36·27-s + 69·29-s + 45·31-s − 54·33-s + 9·35-s − 20·37-s − 15·39-s + 54·41-s + 9·45-s + 207·47-s − 73·49-s − 18·51-s − 252·53-s + 54·55-s − 306·59-s + 7·61-s + ⋯
L(s)  = 1  − 3-s + 3/5·5-s + 3/7·7-s + 1/3·9-s + 1.63·11-s + 5/13·13-s − 3/5·15-s + 6/17·17-s − 3/7·21-s − 3.52·23-s + 1.71·25-s + 4/3·27-s + 2.37·29-s + 1.45·31-s − 1.63·33-s + 9/35·35-s − 0.540·37-s − 0.384·39-s + 1.31·41-s + 1/5·45-s + 4.40·47-s − 1.48·49-s − 0.352·51-s − 4.75·53-s + 0.981·55-s − 5.18·59-s + 7/61·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{16}\)
Sign: $1$
Motivic weight: \(2\)
Character: induced by $\chi_{144} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.02995\)
\(L(\frac12)\) \(\approx\) \(3.02995\)
\(L(2)\) 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 \( 1 + p T + 2 p T^{2} - p^{3} T^{3} - 14 p^{2} T^{4} - p^{5} T^{5} + 2 p^{5} T^{6} + p^{7} T^{7} + p^{8} T^{8} \)
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

−5.82530849102904714351119061198, −5.77055316125661783569707657419, −5.65020988964103856431851396673, −5.07253490157101743692584781237, −4.99350331169179216802146706196, −4.95579028941721166669064100603, −4.82483319495542460263399861402, −4.73323529336662416224810390004, −4.44102966219401460475918758168, −4.29874098430125133403917072557, −4.18049605068638595755532732686, −3.82408886824285930302862034991, −3.64502217604008860925036192930, −3.57716111407378882740140136180, −3.34920815127489152435922556287, −2.88519485308456242758633821767, −2.76542418881860674210321878122, −2.54212287683794237982540762268, −2.52316282509141612422920509808, −1.86023432328296560219928517088, −1.78725362993032460732753634958, −1.33410317274169736428023879096, −1.20968348956690046311347690650, −0.898585450528155957819710722484, −0.35312018984750964854082349129, 0.35312018984750964854082349129, 0.898585450528155957819710722484, 1.20968348956690046311347690650, 1.33410317274169736428023879096, 1.78725362993032460732753634958, 1.86023432328296560219928517088, 2.52316282509141612422920509808, 2.54212287683794237982540762268, 2.76542418881860674210321878122, 2.88519485308456242758633821767, 3.34920815127489152435922556287, 3.57716111407378882740140136180, 3.64502217604008860925036192930, 3.82408886824285930302862034991, 4.18049605068638595755532732686, 4.29874098430125133403917072557, 4.44102966219401460475918758168, 4.73323529336662416224810390004, 4.82483319495542460263399861402, 4.95579028941721166669064100603, 4.99350331169179216802146706196, 5.07253490157101743692584781237, 5.65020988964103856431851396673, 5.77055316125661783569707657419, 5.82530849102904714351119061198

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

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