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Properties

Label 12-4840e6-1.1-c1e6-0-3
Degree $12$
Conductor $1.286\times 10^{22}$
Sign $1$
Analytic cond. $3.33222\times 10^{9}$
Root an. cond. $6.21671$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 6·5-s − 7·7-s − 2·9-s + 13-s + 18·15-s + 6·17-s − 7·19-s + 21·21-s − 9·23-s + 21·25-s + 11·27-s + 10·29-s + 31-s + 42·35-s + 3·37-s − 3·39-s − 6·41-s + 18·43-s + 12·45-s − 3·47-s + 12·49-s − 18·51-s − 23·53-s + 21·57-s − 2·59-s + 6·61-s + ⋯
L(s)  = 1  − 1.73·3-s − 2.68·5-s − 2.64·7-s − 2/3·9-s + 0.277·13-s + 4.64·15-s + 1.45·17-s − 1.60·19-s + 4.58·21-s − 1.87·23-s + 21/5·25-s + 2.11·27-s + 1.85·29-s + 0.179·31-s + 7.09·35-s + 0.493·37-s − 0.480·39-s − 0.937·41-s + 2.74·43-s + 1.78·45-s − 0.437·47-s + 12/7·49-s − 2.52·51-s − 3.15·53-s + 2.78·57-s − 0.260·59-s + 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 5^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(3.33222\times 10^{9}\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{18} \cdot 5^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 + T )^{6} \)
11 \( 1 \)
good3 \( 1 + p T + 11 T^{2} + 28 T^{3} + 67 T^{4} + 137 T^{5} + 253 T^{6} + 137 p T^{7} + 67 p^{2} T^{8} + 28 p^{3} T^{9} + 11 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
7 \( 1 + p T + 37 T^{2} + 152 T^{3} + 582 T^{4} + 37 p^{2} T^{5} + 5144 T^{6} + 37 p^{3} T^{7} + 582 p^{2} T^{8} + 152 p^{3} T^{9} + 37 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
13 \( 1 - T + 45 T^{2} + 6 T^{3} + 70 p T^{4} + 859 T^{5} + 12924 T^{6} + 859 p T^{7} + 70 p^{3} T^{8} + 6 p^{3} T^{9} + 45 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 6 T + 72 T^{2} - 313 T^{3} + 2374 T^{4} - 8118 T^{5} + 48321 T^{6} - 8118 p T^{7} + 2374 p^{2} T^{8} - 313 p^{3} T^{9} + 72 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 7 T + 63 T^{2} + 372 T^{3} + 2629 T^{4} + 11785 T^{5} + 58343 T^{6} + 11785 p T^{7} + 2629 p^{2} T^{8} + 372 p^{3} T^{9} + 63 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 9 T + 90 T^{2} + 583 T^{3} + 3944 T^{4} + 21650 T^{5} + 116494 T^{6} + 21650 p T^{7} + 3944 p^{2} T^{8} + 583 p^{3} T^{9} + 90 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 10 T + 138 T^{2} - 935 T^{3} + 7998 T^{4} - 42085 T^{5} + 281646 T^{6} - 42085 p T^{7} + 7998 p^{2} T^{8} - 935 p^{3} T^{9} + 138 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - T + 91 T^{2} - 256 T^{3} + 4818 T^{4} - 12597 T^{5} + 6032 p T^{6} - 12597 p T^{7} + 4818 p^{2} T^{8} - 256 p^{3} T^{9} + 91 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T + 166 T^{2} - 457 T^{3} + 12968 T^{4} - 30812 T^{5} + 604530 T^{6} - 30812 p T^{7} + 12968 p^{2} T^{8} - 457 p^{3} T^{9} + 166 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 6 T + 160 T^{2} + 865 T^{3} + 12776 T^{4} + 61540 T^{5} + 651329 T^{6} + 61540 p T^{7} + 12776 p^{2} T^{8} + 865 p^{3} T^{9} + 160 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 18 T + 343 T^{2} - 3709 T^{3} + 40406 T^{4} - 309587 T^{5} + 2369267 T^{6} - 309587 p T^{7} + 40406 p^{2} T^{8} - 3709 p^{3} T^{9} + 343 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 3 T + 143 T^{2} + 812 T^{3} + 10526 T^{4} + 74947 T^{5} + 556032 T^{6} + 74947 p T^{7} + 10526 p^{2} T^{8} + 812 p^{3} T^{9} + 143 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 23 T + 7 p T^{2} + 4184 T^{3} + 42556 T^{4} + 368457 T^{5} + 2924216 T^{6} + 368457 p T^{7} + 42556 p^{2} T^{8} + 4184 p^{3} T^{9} + 7 p^{5} T^{10} + 23 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 2 T + 3 p T^{2} - 199 T^{3} + 16248 T^{4} - 37789 T^{5} + 1107507 T^{6} - 37789 p T^{7} + 16248 p^{2} T^{8} - 199 p^{3} T^{9} + 3 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T + 221 T^{2} - 538 T^{3} + 18774 T^{4} + 146 p T^{5} + 1106328 T^{6} + 146 p^{2} T^{7} + 18774 p^{2} T^{8} - 538 p^{3} T^{9} + 221 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 22 T + 298 T^{2} + 3305 T^{3} + 30898 T^{4} + 261640 T^{5} + 2192889 T^{6} + 261640 p T^{7} + 30898 p^{2} T^{8} + 3305 p^{3} T^{9} + 298 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 13 T + 404 T^{2} + 3933 T^{3} + 67568 T^{4} + 509214 T^{5} + 6237970 T^{6} + 509214 p T^{7} + 67568 p^{2} T^{8} + 3933 p^{3} T^{9} + 404 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 10 T + 158 T^{2} - 1819 T^{3} + 25610 T^{4} - 203566 T^{5} + 2001099 T^{6} - 203566 p T^{7} + 25610 p^{2} T^{8} - 1819 p^{3} T^{9} + 158 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 22 T + 506 T^{2} + 7637 T^{3} + 99482 T^{4} + 1123377 T^{5} + 10395066 T^{6} + 1123377 p T^{7} + 99482 p^{2} T^{8} + 7637 p^{3} T^{9} + 506 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 10 T + 313 T^{2} - 2275 T^{3} + 49590 T^{4} - 301275 T^{5} + 5061625 T^{6} - 301275 p T^{7} + 49590 p^{2} T^{8} - 2275 p^{3} T^{9} + 313 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 25 T + 475 T^{2} + 6627 T^{3} + 81130 T^{4} + 837405 T^{5} + 8303699 T^{6} + 837405 p T^{7} + 81130 p^{2} T^{8} + 6627 p^{3} T^{9} + 475 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 33 T + 817 T^{2} + 13611 T^{3} + 195654 T^{4} + 2268901 T^{5} + 24243221 T^{6} + 2268901 p T^{7} + 195654 p^{2} T^{8} + 13611 p^{3} T^{9} + 817 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.82094547741339018345017478913, −4.33924756116459622870721336841, −4.28740148629256773033488794618, −4.25528920295913459558050417592, −3.95346150043281398385445368796, −3.93992682005709714757603832053, −3.91242940979112008240273345846, −3.85735965673954538219557605363, −3.43884069569451286382464956811, −3.36494501002672437682297733214, −3.28588219436335419537645454153, −3.14267724828798814810674030046, −3.12031079262956520240212595510, −2.87930034265349598394531052911, −2.61406507583942826013794687612, −2.56973151991739701686128571983, −2.48626099853669878250226841528, −2.32380628231200034247482796142, −2.25097942854323415561040938841, −1.58226877212548653334479857728, −1.42474196946842174324028120443, −1.36362289718062392240689972498, −1.27216581778177645654091106020, −0.885940779843202223085647412953, −0.853971366402880237445344380379, 0, 0, 0, 0, 0, 0, 0.853971366402880237445344380379, 0.885940779843202223085647412953, 1.27216581778177645654091106020, 1.36362289718062392240689972498, 1.42474196946842174324028120443, 1.58226877212548653334479857728, 2.25097942854323415561040938841, 2.32380628231200034247482796142, 2.48626099853669878250226841528, 2.56973151991739701686128571983, 2.61406507583942826013794687612, 2.87930034265349598394531052911, 3.12031079262956520240212595510, 3.14267724828798814810674030046, 3.28588219436335419537645454153, 3.36494501002672437682297733214, 3.43884069569451286382464956811, 3.85735965673954538219557605363, 3.91242940979112008240273345846, 3.93992682005709714757603832053, 3.95346150043281398385445368796, 4.25528920295913459558050417592, 4.28740148629256773033488794618, 4.33924756116459622870721336841, 4.82094547741339018345017478913

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

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