Properties

Label 12-171e6-1.1-c2e6-0-0
Degree $12$
Conductor $2.500\times 10^{13}$
Sign $1$
Analytic cond. $10232.6$
Root an. cond. $2.15856$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 4-s − 4·5-s − 22·7-s + 6·8-s + 12·10-s + 36·11-s − 3·13-s + 66·14-s − 7·16-s − 38·17-s − 10·19-s − 4·20-s − 108·22-s − 54·23-s + 35·25-s + 9·26-s − 22·28-s + 102·29-s − 21·32-s + 114·34-s + 88·35-s + 30·38-s − 24·40-s − 96·41-s + 107·43-s + 36·44-s + ⋯
L(s)  = 1  − 3/2·2-s + 1/4·4-s − 4/5·5-s − 3.14·7-s + 3/4·8-s + 6/5·10-s + 3.27·11-s − 0.230·13-s + 33/7·14-s − 0.437·16-s − 2.23·17-s − 0.526·19-s − 1/5·20-s − 4.90·22-s − 2.34·23-s + 7/5·25-s + 9/26·26-s − 0.785·28-s + 3.51·29-s − 0.656·32-s + 3.35·34-s + 2.51·35-s + 0.789·38-s − 3/5·40-s − 2.34·41-s + 2.48·43-s + 9/11·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(10232.6\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{171} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 19^{6} ,\ ( \ : [1]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1851758732\)
\(L(\frac12)\) \(\approx\) \(0.1851758732\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + 10 T - 249 T^{2} - 172 p T^{3} - 249 p^{2} T^{4} + 10 p^{4} T^{5} + p^{6} T^{6} \)
good2 \( 1 + 3 T + p^{3} T^{2} + 15 T^{3} + 13 p T^{4} + 57 T^{5} + 71 T^{6} + 57 p^{2} T^{7} + 13 p^{5} T^{8} + 15 p^{6} T^{9} + p^{11} T^{10} + 3 p^{10} T^{11} + p^{12} T^{12} \)
5 \( 1 + 4 T - 19 T^{2} + 84 T^{3} + 238 T^{4} - 704 p T^{5} - 8591 T^{6} - 704 p^{3} T^{7} + 238 p^{4} T^{8} + 84 p^{6} T^{9} - 19 p^{8} T^{10} + 4 p^{10} T^{11} + p^{12} T^{12} \)
7 \( ( 1 + 11 T + 146 T^{2} + 1031 T^{3} + 146 p^{2} T^{4} + 11 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
11 \( ( 1 - 18 T + 267 T^{2} - 3272 T^{3} + 267 p^{2} T^{4} - 18 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
13 \( 1 + 3 T + 167 T^{2} + 492 T^{3} + 101 T^{4} + 65361 T^{5} - 4911130 T^{6} + 65361 p^{2} T^{7} + 101 p^{4} T^{8} + 492 p^{6} T^{9} + 167 p^{8} T^{10} + 3 p^{10} T^{11} + p^{12} T^{12} \)
17 \( 1 + 38 T + 541 T^{2} + 4834 T^{3} - 334 p T^{4} - 3349474 T^{5} - 90714787 T^{6} - 3349474 p^{2} T^{7} - 334 p^{5} T^{8} + 4834 p^{6} T^{9} + 541 p^{8} T^{10} + 38 p^{10} T^{11} + p^{12} T^{12} \)
23 \( 1 + 54 T + 765 T^{2} + 11066 T^{3} + 651546 T^{4} + 5242626 T^{5} - 206650371 T^{6} + 5242626 p^{2} T^{7} + 651546 p^{4} T^{8} + 11066 p^{6} T^{9} + 765 p^{8} T^{10} + 54 p^{10} T^{11} + p^{12} T^{12} \)
29 \( 1 - 102 T + 6203 T^{2} - 278970 T^{3} + 9713378 T^{4} - 292520358 T^{5} + 8574179483 T^{6} - 292520358 p^{2} T^{7} + 9713378 p^{4} T^{8} - 278970 p^{6} T^{9} + 6203 p^{8} T^{10} - 102 p^{10} T^{11} + p^{12} T^{12} \)
31 \( 1 - 3421 T^{2} + 5823134 T^{4} - 6536013745 T^{6} + 5823134 p^{4} T^{8} - 3421 p^{8} T^{10} + p^{12} T^{12} \)
37 \( 1 - 3501 T^{2} + 9576054 T^{4} - 14575249001 T^{6} + 9576054 p^{4} T^{8} - 3501 p^{8} T^{10} + p^{12} T^{12} \)
41 \( 1 + 96 T + 8867 T^{2} + 556320 T^{3} + 34233206 T^{4} + 1610757504 T^{5} + 73428522359 T^{6} + 1610757504 p^{2} T^{7} + 34233206 p^{4} T^{8} + 556320 p^{6} T^{9} + 8867 p^{8} T^{10} + 96 p^{10} T^{11} + p^{12} T^{12} \)
43 \( 1 - 107 T + 2315 T^{2} - 109092 T^{3} + 20090029 T^{4} - 716895265 T^{5} + 7701663934 T^{6} - 716895265 p^{2} T^{7} + 20090029 p^{4} T^{8} - 109092 p^{6} T^{9} + 2315 p^{8} T^{10} - 107 p^{10} T^{11} + p^{12} T^{12} \)
47 \( 1 - 50 T - 4651 T^{2} + 79434 T^{3} + 24415642 T^{4} - 226764442 T^{5} - 55863211187 T^{6} - 226764442 p^{2} T^{7} + 24415642 p^{4} T^{8} + 79434 p^{6} T^{9} - 4651 p^{8} T^{10} - 50 p^{10} T^{11} + p^{12} T^{12} \)
53 \( 1 - 90 T + 123 p T^{2} - 343710 T^{3} + 14892630 T^{4} - 694668222 T^{5} + 20439623059 T^{6} - 694668222 p^{2} T^{7} + 14892630 p^{4} T^{8} - 343710 p^{6} T^{9} + 123 p^{9} T^{10} - 90 p^{10} T^{11} + p^{12} T^{12} \)
59 \( 1 + 9395 T^{2} + 55562030 T^{4} - 82187460 T^{5} + 222918474911 T^{6} - 82187460 p^{2} T^{7} + 55562030 p^{4} T^{8} + 9395 p^{8} T^{10} + p^{12} T^{12} \)
61 \( 1 - 27 T - 8277 T^{2} + 88964 T^{3} + 43113009 T^{4} - 59789481 T^{5} - 181278951354 T^{6} - 59789481 p^{2} T^{7} + 43113009 p^{4} T^{8} + 88964 p^{6} T^{9} - 8277 p^{8} T^{10} - 27 p^{10} T^{11} + p^{12} T^{12} \)
67 \( 1 + 39 T + 7859 T^{2} + 286728 T^{3} + 23155985 T^{4} + 405814017 T^{5} + 60263661254 T^{6} + 405814017 p^{2} T^{7} + 23155985 p^{4} T^{8} + 286728 p^{6} T^{9} + 7859 p^{8} T^{10} + 39 p^{10} T^{11} + p^{12} T^{12} \)
71 \( 1 + 84 T + 10371 T^{2} + 673596 T^{3} + 28816422 T^{4} - 720991884 T^{5} + 18547655047 T^{6} - 720991884 p^{2} T^{7} + 28816422 p^{4} T^{8} + 673596 p^{6} T^{9} + 10371 p^{8} T^{10} + 84 p^{10} T^{11} + p^{12} T^{12} \)
73 \( 1 + 77 T - 1517 T^{2} + 250636 T^{3} + 3339001 T^{4} - 37295617 p T^{5} - 135125227066 T^{6} - 37295617 p^{3} T^{7} + 3339001 p^{4} T^{8} + 250636 p^{6} T^{9} - 1517 p^{8} T^{10} + 77 p^{10} T^{11} + p^{12} T^{12} \)
79 \( 1 - 9 T + 11711 T^{2} - 105156 T^{3} + 64875497 T^{4} - 4632616851 T^{5} + 425401555742 T^{6} - 4632616851 p^{2} T^{7} + 64875497 p^{4} T^{8} - 105156 p^{6} T^{9} + 11711 p^{8} T^{10} - 9 p^{10} T^{11} + p^{12} T^{12} \)
83 \( ( 1 - 174 T + 16527 T^{2} - 1223300 T^{3} + 16527 p^{2} T^{4} - 174 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
89 \( 1 - 72 T + 12639 T^{2} - 785592 T^{3} + 28024818 T^{4} + 6608294460 T^{5} - 147146320001 T^{6} + 6608294460 p^{2} T^{7} + 28024818 p^{4} T^{8} - 785592 p^{6} T^{9} + 12639 p^{8} T^{10} - 72 p^{10} T^{11} + p^{12} T^{12} \)
97 \( 1 + 228 T + 49763 T^{2} + 7395180 T^{3} + 1074017366 T^{4} + 119443328292 T^{5} + 12875825313911 T^{6} + 119443328292 p^{2} T^{7} + 1074017366 p^{4} T^{8} + 7395180 p^{6} T^{9} + 49763 p^{8} T^{10} + 228 p^{10} T^{11} + p^{12} T^{12} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.63114058705045403257828349859, −6.61245850292506738233149908610, −6.55223247387206898285596717931, −6.40167156459776104393174339111, −6.23112506437317849647053572906, −6.11725889634982985596265809278, −5.98589215592205723275501559495, −5.29969493777412332492728414974, −5.22378936050230622680252105843, −4.76985244951054493374735041850, −4.63451036551936690133105561734, −4.51786428977261461144157534958, −4.05993930815258456710308967975, −3.90604710589915128572488345485, −3.85246548139193151892848726746, −3.58763258911429292213201991256, −3.43463996699631144846193794621, −2.90844033227225179242095487131, −2.77423202117855275000312300353, −2.26517308622416227869612254818, −2.22734947434003077963034926984, −1.38719212027469824002866098683, −1.19340417236677942572499899424, −0.53240916620361376618686666712, −0.24161038665417289696026948509, 0.24161038665417289696026948509, 0.53240916620361376618686666712, 1.19340417236677942572499899424, 1.38719212027469824002866098683, 2.22734947434003077963034926984, 2.26517308622416227869612254818, 2.77423202117855275000312300353, 2.90844033227225179242095487131, 3.43463996699631144846193794621, 3.58763258911429292213201991256, 3.85246548139193151892848726746, 3.90604710589915128572488345485, 4.05993930815258456710308967975, 4.51786428977261461144157534958, 4.63451036551936690133105561734, 4.76985244951054493374735041850, 5.22378936050230622680252105843, 5.29969493777412332492728414974, 5.98589215592205723275501559495, 6.11725889634982985596265809278, 6.23112506437317849647053572906, 6.40167156459776104393174339111, 6.55223247387206898285596717931, 6.61245850292506738233149908610, 6.63114058705045403257828349859

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

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