Properties

Label 12-405e6-1.1-c3e6-0-4
Degree $12$
Conductor $4.413\times 10^{15}$
Sign $1$
Analytic cond. $1.86177\times 10^{8}$
Root an. cond. $4.88833$
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  − 2-s + 10·4-s + 15·5-s + 25·7-s − 33·8-s − 15·10-s − 58·11-s + 47·13-s − 25·14-s + 115·16-s + 68·17-s − 10·19-s + 150·20-s + 58·22-s + 51·23-s + 75·25-s − 47·26-s + 250·28-s − 350·29-s − 638·31-s − 414·32-s − 68·34-s + 375·35-s − 828·37-s + 10·38-s − 495·40-s − 179·41-s + ⋯
L(s)  = 1  − 0.353·2-s + 5/4·4-s + 1.34·5-s + 1.34·7-s − 1.45·8-s − 0.474·10-s − 1.58·11-s + 1.00·13-s − 0.477·14-s + 1.79·16-s + 0.970·17-s − 0.120·19-s + 1.67·20-s + 0.562·22-s + 0.462·23-s + 3/5·25-s − 0.354·26-s + 1.68·28-s − 2.24·29-s − 3.69·31-s − 2.28·32-s − 0.342·34-s + 1.81·35-s − 3.67·37-s + 0.0426·38-s − 1.95·40-s − 0.681·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{24} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(1.86177\times 10^{8}\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{405} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{24} \cdot 5^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(3.694338310\)
\(L(\frac12)\) \(\approx\) \(3.694338310\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 - p T + p^{2} T^{2} )^{3} \)
good2 \( 1 + T - 9 T^{2} + 7 p T^{3} + 11 p T^{4} - 29 p^{2} T^{5} + 41 p^{2} T^{6} - 29 p^{5} T^{7} + 11 p^{7} T^{8} + 7 p^{10} T^{9} - 9 p^{12} T^{10} + p^{15} T^{11} + p^{18} T^{12} \)
7 \( 1 - 25 T + 244 T^{2} - 7505 T^{3} - 10772 T^{4} + 3208115 T^{5} - 41593534 T^{6} + 3208115 p^{3} T^{7} - 10772 p^{6} T^{8} - 7505 p^{9} T^{9} + 244 p^{12} T^{10} - 25 p^{15} T^{11} + p^{18} T^{12} \)
11 \( 1 + 58 T + 282 T^{2} - 136036 T^{3} - 3732386 T^{4} + 111996538 T^{5} + 10887063350 T^{6} + 111996538 p^{3} T^{7} - 3732386 p^{6} T^{8} - 136036 p^{9} T^{9} + 282 p^{12} T^{10} + 58 p^{15} T^{11} + p^{18} T^{12} \)
13 \( 1 - 47 T + 3050 T^{2} - 288141 T^{3} + 10255156 T^{4} - 538326055 T^{5} + 35834260864 T^{6} - 538326055 p^{3} T^{7} + 10255156 p^{6} T^{8} - 288141 p^{9} T^{9} + 3050 p^{12} T^{10} - 47 p^{15} T^{11} + p^{18} T^{12} \)
17 \( ( 1 - 2 p T + 12031 T^{2} - 14300 p T^{3} + 12031 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} )^{2} \)
19 \( ( 1 + 5 T + 9800 T^{2} - 231055 T^{3} + 9800 p^{3} T^{4} + 5 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
23 \( 1 - 51 T - 18660 T^{2} - 684555 T^{3} + 183153756 T^{4} + 14586187377 T^{5} - 2242635291686 T^{6} + 14586187377 p^{3} T^{7} + 183153756 p^{6} T^{8} - 684555 p^{9} T^{9} - 18660 p^{12} T^{10} - 51 p^{15} T^{11} + p^{18} T^{12} \)
29 \( 1 + 350 T + 69396 T^{2} + 6917320 T^{3} - 517207640 T^{4} - 388581844690 T^{5} - 78587536986586 T^{6} - 388581844690 p^{3} T^{7} - 517207640 p^{6} T^{8} + 6917320 p^{9} T^{9} + 69396 p^{12} T^{10} + 350 p^{15} T^{11} + p^{18} T^{12} \)
31 \( 1 + 638 T + 182350 T^{2} + 1556452 p T^{3} + 13455116242 T^{4} + 2709597794894 T^{5} + 448785095846702 T^{6} + 2709597794894 p^{3} T^{7} + 13455116242 p^{6} T^{8} + 1556452 p^{10} T^{9} + 182350 p^{12} T^{10} + 638 p^{15} T^{11} + p^{18} T^{12} \)
37 \( ( 1 + 414 T + 167991 T^{2} + 41362924 T^{3} + 167991 p^{3} T^{4} + 414 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
41 \( 1 + 179 T - 22905 T^{2} - 3912848 T^{3} - 1998332639 T^{4} - 127749646003 T^{5} + 341742070644686 T^{6} - 127749646003 p^{3} T^{7} - 1998332639 p^{6} T^{8} - 3912848 p^{9} T^{9} - 22905 p^{12} T^{10} + 179 p^{15} T^{11} + p^{18} T^{12} \)
43 \( 1 - 836 T + 239543 T^{2} - 80761716 T^{3} + 47721809782 T^{4} - 13871341463140 T^{5} + 2802082589374435 T^{6} - 13871341463140 p^{3} T^{7} + 47721809782 p^{6} T^{8} - 80761716 p^{9} T^{9} + 239543 p^{12} T^{10} - 836 p^{15} T^{11} + p^{18} T^{12} \)
47 \( 1 + 5 p T - 186564 T^{2} - 22762453 T^{3} + 24007674184 T^{4} + 355714744651 T^{5} - 2843354974813918 T^{6} + 355714744651 p^{3} T^{7} + 24007674184 p^{6} T^{8} - 22762453 p^{9} T^{9} - 186564 p^{12} T^{10} + 5 p^{16} T^{11} + p^{18} T^{12} \)
53 \( ( 1 - 505 T + 481759 T^{2} - 148865086 T^{3} + 481759 p^{3} T^{4} - 505 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
59 \( 1 + 535 T - 78663 T^{2} - 288931294 T^{3} - 71306235053 T^{4} + 30710030725351 T^{5} + 36507468196017446 T^{6} + 30710030725351 p^{3} T^{7} - 71306235053 p^{6} T^{8} - 288931294 p^{9} T^{9} - 78663 p^{12} T^{10} + 535 p^{15} T^{11} + p^{18} T^{12} \)
61 \( 1 - 104 T - 375715 T^{2} + 7139208 T^{3} + 59209422586 T^{4} + 3741915462248 T^{5} - 10521092248447379 T^{6} + 3741915462248 p^{3} T^{7} + 59209422586 p^{6} T^{8} + 7139208 p^{9} T^{9} - 375715 p^{12} T^{10} - 104 p^{15} T^{11} + p^{18} T^{12} \)
67 \( 1 - 40 T - 525509 T^{2} - 4394696 T^{3} + 118975220434 T^{4} + 4665819395912 T^{5} - 29227192656155485 T^{6} + 4665819395912 p^{3} T^{7} + 118975220434 p^{6} T^{8} - 4394696 p^{9} T^{9} - 525509 p^{12} T^{10} - 40 p^{15} T^{11} + p^{18} T^{12} \)
71 \( ( 1 - 452 T + 808810 T^{2} - 207368090 T^{3} + 808810 p^{3} T^{4} - 452 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
73 \( ( 1 + 710 T + 1311287 T^{2} + 561111668 T^{3} + 1311287 p^{3} T^{4} + 710 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
79 \( 1 - 634 T - 956873 T^{2} + 398955430 T^{3} + 776896953838 T^{4} - 160983685831738 T^{5} - 371177040015899773 T^{6} - 160983685831738 p^{3} T^{7} + 776896953838 p^{6} T^{8} + 398955430 p^{9} T^{9} - 956873 p^{12} T^{10} - 634 p^{15} T^{11} + p^{18} T^{12} \)
83 \( 1 + 1734 T + 1009443 T^{2} - 57904194 T^{3} - 205697747874 T^{4} + 122842325158422 T^{5} + 209635494483596263 T^{6} + 122842325158422 p^{3} T^{7} - 205697747874 p^{6} T^{8} - 57904194 p^{9} T^{9} + 1009443 p^{12} T^{10} + 1734 p^{15} T^{11} + p^{18} T^{12} \)
89 \( ( 1 + 852 T + 2335632 T^{2} + 1219193610 T^{3} + 2335632 p^{3} T^{4} + 852 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
97 \( 1 - 1162851 T^{2} - 1406550016 T^{3} + 290919737478 T^{4} + 817804046327808 T^{5} + 780920935719579225 T^{6} + 817804046327808 p^{3} T^{7} + 290919737478 p^{6} T^{8} - 1406550016 p^{9} T^{9} - 1162851 p^{12} T^{10} + p^{18} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.45589062313432219571450505213, −5.43317574337670770859854661888, −5.36410453742457851338419371288, −5.35260601465472335097228539087, −5.24408411251321696149708634830, −4.71387589198935980207559107384, −4.63721629702061447247842905307, −4.29262516568776356789584558125, −4.15396517368751670373484120546, −3.70208421303497943931356888372, −3.53019946053603110106015213336, −3.44224426983657987256786102788, −3.34114668211825418911553238893, −3.14814590930375470168181931129, −2.97379100117213376636589282787, −2.22382195018894353396034697457, −2.21028490583025403186974680587, −2.11844280633748984754206853028, −2.09318911140121576853238322314, −1.84965309016722036436336019522, −1.43178698246530459002536021264, −1.13061778395147925968531348964, −1.06687844703190379614673497987, −0.39970941342470418892747316575, −0.21093754209155160538193839883, 0.21093754209155160538193839883, 0.39970941342470418892747316575, 1.06687844703190379614673497987, 1.13061778395147925968531348964, 1.43178698246530459002536021264, 1.84965309016722036436336019522, 2.09318911140121576853238322314, 2.11844280633748984754206853028, 2.21028490583025403186974680587, 2.22382195018894353396034697457, 2.97379100117213376636589282787, 3.14814590930375470168181931129, 3.34114668211825418911553238893, 3.44224426983657987256786102788, 3.53019946053603110106015213336, 3.70208421303497943931356888372, 4.15396517368751670373484120546, 4.29262516568776356789584558125, 4.63721629702061447247842905307, 4.71387589198935980207559107384, 5.24408411251321696149708634830, 5.35260601465472335097228539087, 5.36410453742457851338419371288, 5.43317574337670770859854661888, 5.45589062313432219571450505213

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

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