Properties

Label 16-304e8-1.1-c6e8-0-0
Degree $16$
Conductor $7.294\times 10^{19}$
Sign $1$
Analytic cond. $5.72305\times 10^{14}$
Root an. cond. $8.36280$
Motivic weight $6$
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  + 2·5-s − 362·7-s + 742·9-s − 902·11-s + 1.55e3·17-s − 6.23e3·19-s + 1.88e4·23-s − 6.85e4·25-s − 724·35-s + 3.35e5·43-s + 1.48e3·45-s + 5.70e5·47-s − 1.80e5·49-s − 1.80e3·55-s − 6.32e5·61-s − 2.68e5·63-s − 8.52e5·73-s + 3.26e5·77-s − 9.72e5·81-s − 4.41e5·83-s + 3.10e3·85-s − 1.24e4·95-s − 6.69e5·99-s − 1.81e6·101-s + 3.76e4·115-s − 5.61e5·119-s − 5.29e6·121-s + ⋯
L(s)  = 1  + 0.0159·5-s − 1.05·7-s + 1.01·9-s − 0.677·11-s + 0.315·17-s − 0.908·19-s + 1.54·23-s − 4.38·25-s − 0.0168·35-s + 4.21·43-s + 0.0162·45-s + 5.49·47-s − 1.53·49-s − 0.0108·55-s − 2.78·61-s − 1.07·63-s − 2.19·73-s + 0.715·77-s − 1.82·81-s − 0.771·83-s + 0.00504·85-s − 0.0145·95-s − 0.689·99-s − 1.76·101-s + 0.0247·115-s − 0.332·119-s − 2.98·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(5.72305\times 10^{14}\)
Root analytic conductor: \(8.36280\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 19^{8} ,\ ( \ : [3]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.001809907527\)
\(L(\frac12)\) \(\approx\) \(0.001809907527\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + 328 p T + 7370512 p T^{2} + 1950884072 p^{2} T^{3} + 70846142174 p^{4} T^{4} + 1950884072 p^{8} T^{5} + 7370512 p^{13} T^{6} + 328 p^{19} T^{7} + p^{24} T^{8} \)
good3 \( 1 - 742 T^{2} + 507523 p T^{4} - 139764998 p^{2} T^{6} + 39023898956 p^{3} T^{8} - 139764998 p^{14} T^{10} + 507523 p^{25} T^{12} - 742 p^{36} T^{14} + p^{48} T^{16} \)
5 \( ( 1 - T + 6858 p T^{2} + 57133 p^{2} T^{3} + 4953194 p^{3} T^{4} + 57133 p^{8} T^{5} + 6858 p^{13} T^{6} - p^{18} T^{7} + p^{24} T^{8} )^{2} \)
7 \( ( 1 + 181 T + 139633 T^{2} - 29351086 T^{3} - 320349574 T^{4} - 29351086 p^{6} T^{5} + 139633 p^{12} T^{6} + 181 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
11 \( ( 1 + 41 p T + 2951358 T^{2} + 3849343493 T^{3} + 4110119555458 T^{4} + 3849343493 p^{6} T^{5} + 2951358 p^{12} T^{6} + 41 p^{19} T^{7} + p^{24} T^{8} )^{2} \)
13 \( 1 - 18101774 T^{2} + 149249945971081 T^{4} - \)\(86\!\cdots\!26\)\( T^{6} + \)\(43\!\cdots\!16\)\( T^{8} - \)\(86\!\cdots\!26\)\( p^{12} T^{10} + 149249945971081 p^{24} T^{12} - 18101774 p^{36} T^{14} + p^{48} T^{16} \)
17 \( ( 1 - 775 T + 76306401 T^{2} - 62323644578 T^{3} + 2574852969478894 T^{4} - 62323644578 p^{6} T^{5} + 76306401 p^{12} T^{6} - 775 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
23 \( ( 1 - 9410 T + 537866745 T^{2} - 3615134698390 T^{3} + 115088224709036668 T^{4} - 3615134698390 p^{6} T^{5} + 537866745 p^{12} T^{6} - 9410 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
29 \( 1 - 1955972438 T^{2} + 1491101576657216569 T^{4} - \)\(39\!\cdots\!02\)\( T^{6} - \)\(19\!\cdots\!00\)\( T^{8} - \)\(39\!\cdots\!02\)\( p^{12} T^{10} + 1491101576657216569 p^{24} T^{12} - 1955972438 p^{36} T^{14} + p^{48} T^{16} \)
31 \( 1 - 1360240952 T^{2} + 2738159624622996748 T^{4} - \)\(21\!\cdots\!68\)\( T^{6} + \)\(27\!\cdots\!82\)\( T^{8} - \)\(21\!\cdots\!68\)\( p^{12} T^{10} + 2738159624622996748 p^{24} T^{12} - 1360240952 p^{36} T^{14} + p^{48} T^{16} \)
37 \( 1 - 7244329016 T^{2} + 23581771984582865740 T^{4} - \)\(37\!\cdots\!88\)\( T^{6} + \)\(48\!\cdots\!10\)\( T^{8} - \)\(37\!\cdots\!88\)\( p^{12} T^{10} + 23581771984582865740 p^{24} T^{12} - 7244329016 p^{36} T^{14} + p^{48} T^{16} \)
41 \( 1 - 4812538424 T^{2} + 68132340249279236236 T^{4} - \)\(21\!\cdots\!16\)\( T^{6} + \)\(48\!\cdots\!26\)\( p T^{8} - \)\(21\!\cdots\!16\)\( p^{12} T^{10} + 68132340249279236236 p^{24} T^{12} - 4812538424 p^{36} T^{14} + p^{48} T^{16} \)
43 \( ( 1 - 167521 T + 28059854686 T^{2} - 2860511677640443 T^{3} + \)\(27\!\cdots\!70\)\( T^{4} - 2860511677640443 p^{6} T^{5} + 28059854686 p^{12} T^{6} - 167521 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
47 \( ( 1 - 285197 T + 53014565646 T^{2} - 6790341576595363 T^{3} + \)\(75\!\cdots\!74\)\( T^{4} - 6790341576595363 p^{6} T^{5} + 53014565646 p^{12} T^{6} - 285197 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
53 \( 1 - 106845552662 T^{2} + \)\(57\!\cdots\!21\)\( T^{4} - \)\(20\!\cdots\!82\)\( T^{6} + \)\(53\!\cdots\!08\)\( T^{8} - \)\(20\!\cdots\!82\)\( p^{12} T^{10} + \)\(57\!\cdots\!21\)\( p^{24} T^{12} - 106845552662 p^{36} T^{14} + p^{48} T^{16} \)
59 \( 1 - 283261383806 T^{2} + \)\(36\!\cdots\!69\)\( T^{4} - \)\(28\!\cdots\!06\)\( T^{6} + \)\(14\!\cdots\!44\)\( T^{8} - \)\(28\!\cdots\!06\)\( p^{12} T^{10} + \)\(36\!\cdots\!69\)\( p^{24} T^{12} - 283261383806 p^{36} T^{14} + p^{48} T^{16} \)
61 \( ( 1 + 316007 T + 182153576986 T^{2} + 45019267728697357 T^{3} + \)\(13\!\cdots\!70\)\( T^{4} + 45019267728697357 p^{6} T^{5} + 182153576986 p^{12} T^{6} + 316007 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
67 \( 1 - 284119902326 T^{2} + \)\(40\!\cdots\!01\)\( T^{4} - \)\(44\!\cdots\!94\)\( T^{6} + \)\(42\!\cdots\!76\)\( T^{8} - \)\(44\!\cdots\!94\)\( p^{12} T^{10} + \)\(40\!\cdots\!01\)\( p^{24} T^{12} - 284119902326 p^{36} T^{14} + p^{48} T^{16} \)
71 \( 1 - 550447281824 T^{2} + \)\(16\!\cdots\!32\)\( T^{4} - \)\(34\!\cdots\!36\)\( T^{6} + \)\(52\!\cdots\!54\)\( T^{8} - \)\(34\!\cdots\!36\)\( p^{12} T^{10} + \)\(16\!\cdots\!32\)\( p^{24} T^{12} - 550447281824 p^{36} T^{14} + p^{48} T^{16} \)
73 \( ( 1 + 426469 T + 412154349109 T^{2} + 175844843253819082 T^{3} + \)\(80\!\cdots\!22\)\( T^{4} + 175844843253819082 p^{6} T^{5} + 412154349109 p^{12} T^{6} + 426469 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
79 \( 1 - 935745384632 T^{2} + \)\(48\!\cdots\!64\)\( T^{4} - \)\(17\!\cdots\!52\)\( T^{6} + \)\(48\!\cdots\!78\)\( T^{8} - \)\(17\!\cdots\!52\)\( p^{12} T^{10} + \)\(48\!\cdots\!64\)\( p^{24} T^{12} - 935745384632 p^{36} T^{14} + p^{48} T^{16} \)
83 \( ( 1 + 220600 T + 1020667843728 T^{2} + 182031252335003816 T^{3} + \)\(45\!\cdots\!30\)\( T^{4} + 182031252335003816 p^{6} T^{5} + 1020667843728 p^{12} T^{6} + 220600 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
89 \( 1 - 1390950826016 T^{2} + \)\(76\!\cdots\!16\)\( T^{4} - \)\(73\!\cdots\!44\)\( T^{6} - \)\(68\!\cdots\!94\)\( T^{8} - \)\(73\!\cdots\!44\)\( p^{12} T^{10} + \)\(76\!\cdots\!16\)\( p^{24} T^{12} - 1390950826016 p^{36} T^{14} + p^{48} T^{16} \)
97 \( 1 - 5125637527808 T^{2} + \)\(12\!\cdots\!48\)\( T^{4} - \)\(18\!\cdots\!72\)\( T^{6} + \)\(18\!\cdots\!02\)\( T^{8} - \)\(18\!\cdots\!72\)\( p^{12} T^{10} + \)\(12\!\cdots\!48\)\( p^{24} T^{12} - 5125637527808 p^{36} T^{14} + p^{48} T^{16} \)
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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

−4.19406608367654231147673758780, −4.07911109695345531751842579822, −3.94137889401728045138329344196, −3.90273605606814934935855696292, −3.32426842563673912247390678876, −3.26065901609252338395042729524, −3.19621504426441021176783101625, −3.02954303404379845264919463108, −3.02901169125959871083188973386, −2.70020522590482620596653464303, −2.52481284045350325382828233683, −2.36878492094067651821222798700, −2.20935507112322165355396735383, −2.19048956725235871012191104231, −1.86401817690244553997929275535, −1.70362945589708948622108248513, −1.55007785210807606773144178328, −1.38363723556910175189505193106, −1.26166886921367116336565412940, −0.932636827945121013616168458251, −0.69617938208481102212027228060, −0.63679406539094579183016351897, −0.58366868157076524199053447920, −0.18635819042941145036396823766, −0.00390699275237358958457974104, 0.00390699275237358958457974104, 0.18635819042941145036396823766, 0.58366868157076524199053447920, 0.63679406539094579183016351897, 0.69617938208481102212027228060, 0.932636827945121013616168458251, 1.26166886921367116336565412940, 1.38363723556910175189505193106, 1.55007785210807606773144178328, 1.70362945589708948622108248513, 1.86401817690244553997929275535, 2.19048956725235871012191104231, 2.20935507112322165355396735383, 2.36878492094067651821222798700, 2.52481284045350325382828233683, 2.70020522590482620596653464303, 3.02901169125959871083188973386, 3.02954303404379845264919463108, 3.19621504426441021176783101625, 3.26065901609252338395042729524, 3.32426842563673912247390678876, 3.90273605606814934935855696292, 3.94137889401728045138329344196, 4.07911109695345531751842579822, 4.19406608367654231147673758780

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

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