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Properties

Label 12-2e24-1.1-c2e6-0-0
Degree $12$
Conductor $16777216$
Sign $1$
Analytic cond. $0.00686641$
Root an. cond. $0.660279$
Motivic weight $2$
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  − 2·2-s − 2·3-s − 2·4-s − 2·5-s + 4·6-s − 4·7-s + 8·8-s + 2·9-s + 4·10-s − 18·11-s + 4·12-s − 2·13-s + 8·14-s + 4·15-s − 12·16-s − 4·17-s − 4·18-s + 30·19-s + 4·20-s + 8·21-s + 36·22-s + 60·23-s − 16·24-s + 2·25-s + 4·26-s + 14·27-s + 8·28-s + ⋯
L(s)  = 1  − 2-s − 2/3·3-s − 1/2·4-s − 2/5·5-s + 2/3·6-s − 4/7·7-s + 8-s + 2/9·9-s + 2/5·10-s − 1.63·11-s + 1/3·12-s − 0.153·13-s + 4/7·14-s + 4/15·15-s − 3/4·16-s − 0.235·17-s − 2/9·18-s + 1.57·19-s + 1/5·20-s + 8/21·21-s + 1.63·22-s + 2.60·23-s − 2/3·24-s + 2/25·25-s + 2/13·26-s + 0.518·27-s + 2/7·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16777216 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16777216 ^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(16777216\)    =    \(2^{24}\)
Sign: $1$
Analytic conductor: \(0.00686641\)
Root analytic conductor: \(0.660279\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 16777216,\ (\ :[1]^{6}),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1229477612\)
\(L(\frac12)\) \(\approx\) \(0.1229477612\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T + 3 p T^{2} + p^{3} T^{3} + 3 p^{3} T^{4} + p^{5} T^{5} + p^{6} T^{6} \)
good3 \( 1 + 2 T + 2 T^{2} - 14 T^{3} - 65 T^{4} + 124 T^{5} + 476 T^{6} + 124 p^{2} T^{7} - 65 p^{4} T^{8} - 14 p^{6} T^{9} + 2 p^{8} T^{10} + 2 p^{10} T^{11} + p^{12} T^{12} \)
5 \( 1 + 2 T + 2 T^{2} - 14 T^{3} - 369 T^{4} + 636 T^{5} + 2108 T^{6} + 636 p^{2} T^{7} - 369 p^{4} T^{8} - 14 p^{6} T^{9} + 2 p^{8} T^{10} + 2 p^{10} T^{11} + p^{12} T^{12} \)
7 \( ( 1 + 2 T + 87 T^{2} + 332 T^{3} + 87 p^{2} T^{4} + 2 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
11 \( 1 + 18 T + 162 T^{2} + 2146 T^{3} + 17759 T^{4} + 65756 T^{5} + 609308 T^{6} + 65756 p^{2} T^{7} + 17759 p^{4} T^{8} + 2146 p^{6} T^{9} + 162 p^{8} T^{10} + 18 p^{10} T^{11} + p^{12} T^{12} \)
13 \( 1 + 2 T + 2 T^{2} + 1554 T^{3} - 7825 T^{4} - 453380 T^{5} + 316348 T^{6} - 453380 p^{2} T^{7} - 7825 p^{4} T^{8} + 1554 p^{6} T^{9} + 2 p^{8} T^{10} + 2 p^{10} T^{11} + p^{12} T^{12} \)
17 \( ( 1 + 2 T + 607 T^{2} - 388 T^{3} + 607 p^{2} T^{4} + 2 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
19 \( 1 - 30 T + 450 T^{2} - 12014 T^{3} + 441215 T^{4} - 8004292 T^{5} + 113750108 T^{6} - 8004292 p^{2} T^{7} + 441215 p^{4} T^{8} - 12014 p^{6} T^{9} + 450 p^{8} T^{10} - 30 p^{10} T^{11} + p^{12} T^{12} \)
23 \( ( 1 - 30 T + 1751 T^{2} - 30772 T^{3} + 1751 p^{2} T^{4} - 30 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
29 \( 1 + 18 T + 162 T^{2} - 4894 T^{3} + 124463 T^{4} + 24625372 T^{5} + 435069308 T^{6} + 24625372 p^{2} T^{7} + 124463 p^{4} T^{8} - 4894 p^{6} T^{9} + 162 p^{8} T^{10} + 18 p^{10} T^{11} + p^{12} T^{12} \)
31 \( 1 - 3846 T^{2} + 7131791 T^{4} - 8361808916 T^{6} + 7131791 p^{4} T^{8} - 3846 p^{8} T^{10} + p^{12} T^{12} \)
37 \( 1 - 46 T + 1058 T^{2} + 6594 T^{3} - 356337 T^{4} - 78343460 T^{5} + 4002544124 T^{6} - 78343460 p^{2} T^{7} - 356337 p^{4} T^{8} + 6594 p^{6} T^{9} + 1058 p^{8} T^{10} - 46 p^{10} T^{11} + p^{12} T^{12} \)
41 \( 1 - 5094 T^{2} + 15050223 T^{4} - 31243096276 T^{6} + 15050223 p^{4} T^{8} - 5094 p^{8} T^{10} + p^{12} T^{12} \)
43 \( 1 + 114 T + 6498 T^{2} + 241730 T^{3} + 12357983 T^{4} + 838941724 T^{5} + 44553879452 T^{6} + 838941724 p^{2} T^{7} + 12357983 p^{4} T^{8} + 241730 p^{6} T^{9} + 6498 p^{8} T^{10} + 114 p^{10} T^{11} + p^{12} T^{12} \)
47 \( 1 - 4678 T^{2} + 12462287 T^{4} - 24905944212 T^{6} + 12462287 p^{4} T^{8} - 4678 p^{8} T^{10} + p^{12} T^{12} \)
53 \( 1 - 78 T + 3042 T^{2} - 270110 T^{3} + 31648463 T^{4} - 1389102820 T^{5} + 48555101564 T^{6} - 1389102820 p^{2} T^{7} + 31648463 p^{4} T^{8} - 270110 p^{6} T^{9} + 3042 p^{8} T^{10} - 78 p^{10} T^{11} + p^{12} T^{12} \)
59 \( 1 - 206 T + 21218 T^{2} - 1942462 T^{3} + 171214239 T^{4} - 11916831972 T^{5} + 708622973852 T^{6} - 11916831972 p^{2} T^{7} + 171214239 p^{4} T^{8} - 1942462 p^{6} T^{9} + 21218 p^{8} T^{10} - 206 p^{10} T^{11} + p^{12} T^{12} \)
61 \( 1 - 30 T + 450 T^{2} - 111694 T^{3} + 33268655 T^{4} - 582006980 T^{5} + 8727089468 T^{6} - 582006980 p^{2} T^{7} + 33268655 p^{4} T^{8} - 111694 p^{6} T^{9} + 450 p^{8} T^{10} - 30 p^{10} T^{11} + p^{12} T^{12} \)
67 \( 1 + 226 T + 25538 T^{2} + 2083538 T^{3} + 120508479 T^{4} + 5203289532 T^{5} + 268963196252 T^{6} + 5203289532 p^{2} T^{7} + 120508479 p^{4} T^{8} + 2083538 p^{6} T^{9} + 25538 p^{8} T^{10} + 226 p^{10} T^{11} + p^{12} T^{12} \)
71 \( ( 1 + 130 T + 11575 T^{2} + 918796 T^{3} + 11575 p^{2} T^{4} + 130 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
73 \( 1 - 13126 T^{2} + 103571951 T^{4} - 653716749588 T^{6} + 103571951 p^{4} T^{8} - 13126 p^{8} T^{10} + p^{12} T^{12} \)
79 \( 1 - 70 T^{2} + 84324175 T^{4} + 17226941804 T^{6} + 84324175 p^{4} T^{8} - 70 p^{8} T^{10} + p^{12} T^{12} \)
83 \( 1 - 318 T + 50562 T^{2} - 6712846 T^{3} + 819490815 T^{4} - 81203275140 T^{5} + 6918697616348 T^{6} - 81203275140 p^{2} T^{7} + 819490815 p^{4} T^{8} - 6712846 p^{6} T^{9} + 50562 p^{8} T^{10} - 318 p^{10} T^{11} + p^{12} T^{12} \)
89 \( 1 - 31238 T^{2} + 466178479 T^{4} - 4433595811988 T^{6} + 466178479 p^{4} T^{8} - 31238 p^{8} T^{10} + p^{12} T^{12} \)
97 \( ( 1 + 2 T + 10687 T^{2} + 557564 T^{3} + 10687 p^{2} T^{4} + 2 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
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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

−11.36070723735327047177036977865, −11.15029139372647895587989645814, −11.01739937503836028067112955293, −10.43946517157300583994667948221, −10.14434279059121646655929681001, −10.09800024692954469366961831329, −10.06652148021096439654084648890, −9.327656388993218458874962249077, −9.230626755154415845242817721063, −8.994231981046499986081599454075, −8.811955731886072562243428547535, −8.200020929716778120328955933920, −8.129835113095462546847915139108, −7.66736938703655029869243953012, −7.49795545202933616860365622989, −7.00661769518024865851982438880, −6.71654148449982213536092582970, −6.53005724998402159368645936403, −5.77183978837161210226899607114, −5.23955131995947565398025839100, −5.23075575454201167238505510963, −4.75617402699940195329251308925, −4.30895712567018791429387203306, −3.26337001318846697709200002217, −2.98931204697392862733797842527, 2.98931204697392862733797842527, 3.26337001318846697709200002217, 4.30895712567018791429387203306, 4.75617402699940195329251308925, 5.23075575454201167238505510963, 5.23955131995947565398025839100, 5.77183978837161210226899607114, 6.53005724998402159368645936403, 6.71654148449982213536092582970, 7.00661769518024865851982438880, 7.49795545202933616860365622989, 7.66736938703655029869243953012, 8.129835113095462546847915139108, 8.200020929716778120328955933920, 8.811955731886072562243428547535, 8.994231981046499986081599454075, 9.230626755154415845242817721063, 9.327656388993218458874962249077, 10.06652148021096439654084648890, 10.09800024692954469366961831329, 10.14434279059121646655929681001, 10.43946517157300583994667948221, 11.01739937503836028067112955293, 11.15029139372647895587989645814, 11.36070723735327047177036977865

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

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