Dirichlet series
| L(s) = 1 | + 6·2-s + 9·3-s + 12·4-s − 2·5-s + 54·6-s − 34·7-s − 16·8-s + 27·9-s − 12·10-s − 104·11-s + 108·12-s − 75·13-s − 204·14-s − 18·15-s − 144·16-s + 48·17-s + 162·18-s + 104·19-s − 24·20-s − 306·21-s − 624·22-s + 238·23-s − 144·24-s + 75·25-s − 450·26-s − 54·27-s − 408·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3/2·4-s − 0.178·5-s + 3.67·6-s − 1.83·7-s − 0.707·8-s + 9-s − 0.379·10-s − 2.85·11-s + 2.59·12-s − 1.60·13-s − 3.89·14-s − 0.309·15-s − 9/4·16-s + 0.684·17-s + 2.12·18-s + 1.25·19-s − 0.268·20-s − 3.17·21-s − 6.04·22-s + 2.15·23-s − 1.22·24-s + 3/5·25-s − 3.39·26-s − 0.384·27-s − 2.75·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{6} \cdot 19^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(92603.1\) |
| Root analytic conductor: | \(2.59349\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.1182694049\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1182694049\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T + p^{2} T^{2} )^{3} \) |
| 3 | \( ( 1 - p T + p^{2} T^{2} )^{3} \) | |
| 19 | \( 1 - 104 T + 443 p T^{2} - 3056 p^{2} T^{3} + 443 p^{4} T^{4} - 104 p^{6} T^{5} + p^{9} T^{6} \) | |
| good | 5 | \( 1 + 2 T - 71 T^{2} - 3586 T^{3} - 7486 T^{4} + 5354 p^{2} T^{5} + 6630049 T^{6} + 5354 p^{5} T^{7} - 7486 p^{6} T^{8} - 3586 p^{9} T^{9} - 71 p^{12} T^{10} + 2 p^{15} T^{11} + p^{18} T^{12} \) |
| 7 | \( ( 1 + 17 T + 88 T^{2} - 6791 T^{3} + 88 p^{3} T^{4} + 17 p^{6} T^{5} + p^{9} T^{6} )^{2} \) | |
| 11 | \( ( 1 + 52 T + 3105 T^{2} + 105736 T^{3} + 3105 p^{3} T^{4} + 52 p^{6} T^{5} + p^{9} T^{6} )^{2} \) | |
| 13 | \( 1 + 75 T - 1785 T^{2} - 76936 T^{3} + 14902605 T^{4} + 20318985 p T^{5} - 26502003930 T^{6} + 20318985 p^{4} T^{7} + 14902605 p^{6} T^{8} - 76936 p^{9} T^{9} - 1785 p^{12} T^{10} + 75 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 - 48 T - 10707 T^{2} + 339504 T^{3} + 82226310 T^{4} - 1293501360 T^{5} - 417800494199 T^{6} - 1293501360 p^{3} T^{7} + 82226310 p^{6} T^{8} + 339504 p^{9} T^{9} - 10707 p^{12} T^{10} - 48 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 - 238 T + 43195 T^{2} - 3827206 T^{3} + 91776554 T^{4} + 46869084554 T^{5} - 6885676309373 T^{6} + 46869084554 p^{3} T^{7} + 91776554 p^{6} T^{8} - 3827206 p^{9} T^{9} + 43195 p^{12} T^{10} - 238 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 - 8 T - 30287 T^{2} - 75224 T^{3} + 180282326 T^{4} + 4541931064 T^{5} + 6516507269269 T^{6} + 4541931064 p^{3} T^{7} + 180282326 p^{6} T^{8} - 75224 p^{9} T^{9} - 30287 p^{12} T^{10} - 8 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( ( 1 - 107 T + 75368 T^{2} - 4940027 T^{3} + 75368 p^{3} T^{4} - 107 p^{6} T^{5} + p^{9} T^{6} )^{2} \) | |
| 37 | \( ( 1 - 305 T + 26786 T^{2} - 1997981 T^{3} + 26786 p^{3} T^{4} - 305 p^{6} T^{5} + p^{9} T^{6} )^{2} \) | |
| 41 | \( 1 + 16 T - 39947 T^{2} - 17782928 T^{3} - 1301959162 T^{4} + 357728853904 T^{5} + 627033115392481 T^{6} + 357728853904 p^{3} T^{7} - 1301959162 p^{6} T^{8} - 17782928 p^{9} T^{9} - 39947 p^{12} T^{10} + 16 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 - 331 T - 45019 T^{2} + 60344490 T^{3} - 6850198445 T^{4} - 2574908251799 T^{5} + 1521438708132622 T^{6} - 2574908251799 p^{3} T^{7} - 6850198445 p^{6} T^{8} + 60344490 p^{9} T^{9} - 45019 p^{12} T^{10} - 331 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 - 766 T + 183979 T^{2} - 30806518 T^{3} + 14044780946 T^{4} - 122191650406 T^{5} - 1647450351866933 T^{6} - 122191650406 p^{3} T^{7} + 14044780946 p^{6} T^{8} - 30806518 p^{9} T^{9} + 183979 p^{12} T^{10} - 766 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 - 118 T - 214727 T^{2} - 38298826 T^{3} + 18908167346 T^{4} + 6712188793154 T^{5} - 1460545147417391 T^{6} + 6712188793154 p^{3} T^{7} + 18908167346 p^{6} T^{8} - 38298826 p^{9} T^{9} - 214727 p^{12} T^{10} - 118 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 936 T + 109839 T^{2} + 11616552 T^{3} + 99555893646 T^{4} + 15214012621560 T^{5} - 15748601893909229 T^{6} + 15214012621560 p^{3} T^{7} + 99555893646 p^{6} T^{8} + 11616552 p^{9} T^{9} + 109839 p^{12} T^{10} + 936 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 399 T - 312153 T^{2} + 17847104 T^{3} + 74105679717 T^{4} + 16290132348663 T^{5} - 24642725540680650 T^{6} + 16290132348663 p^{3} T^{7} + 74105679717 p^{6} T^{8} + 17847104 p^{9} T^{9} - 312153 p^{12} T^{10} - 399 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 + 61 T - 710307 T^{2} - 85042350 T^{3} + 291160540651 T^{4} + 376557874339 p T^{5} - 21191570953154 p^{2} T^{6} + 376557874339 p^{4} T^{7} + 291160540651 p^{6} T^{8} - 85042350 p^{9} T^{9} - 710307 p^{12} T^{10} + 61 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 974 T - 146477 T^{2} - 293762410 T^{3} + 144857810642 T^{4} + 106163123442518 T^{5} + 3607388571303523 T^{6} + 106163123442518 p^{3} T^{7} + 144857810642 p^{6} T^{8} - 293762410 p^{9} T^{9} - 146477 p^{12} T^{10} + 974 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 + 91 T - 590469 T^{2} + 247304940 T^{3} + 134350884145 T^{4} - 86345835758951 T^{5} - 13678465847096858 T^{6} - 86345835758951 p^{3} T^{7} + 134350884145 p^{6} T^{8} + 247304940 p^{9} T^{9} - 590469 p^{12} T^{10} + 91 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 - 321 T - 948495 T^{2} + 422984066 T^{3} + 465215098311 T^{4} - 145048115527569 T^{5} - 185911414038042546 T^{6} - 145048115527569 p^{3} T^{7} + 465215098311 p^{6} T^{8} + 422984066 p^{9} T^{9} - 948495 p^{12} T^{10} - 321 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( ( 1 + 2148 T + 2987025 T^{2} + 2667812568 T^{3} + 2987025 p^{3} T^{4} + 2148 p^{6} T^{5} + p^{9} T^{6} )^{2} \) | |
| 89 | \( 1 + 1116 T - 1114611 T^{2} - 535222908 T^{3} + 2137807986126 T^{4} + 580681615780620 T^{5} - 1334142163469573039 T^{6} + 580681615780620 p^{3} T^{7} + 2137807986126 p^{6} T^{8} - 535222908 p^{9} T^{9} - 1114611 p^{12} T^{10} + 1116 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 + 1382 T - 1429195 T^{2} - 589150062 T^{3} + 4506357489754 T^{4} + 1415886280736350 T^{5} - 3476579393968045715 T^{6} + 1415886280736350 p^{3} T^{7} + 4506357489754 p^{6} T^{8} - 589150062 p^{9} T^{9} - 1429195 p^{12} T^{10} + 1382 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−7.07794961465135876311224205509, −6.85915770406683865314038358087, −6.77670778135530802852219123197, −6.55365354972886238446239958690, −5.93387226490718014826913399535, −5.81889782441868448295012242101, −5.56109817647016539174293206730, −5.54925073888602436600916379674, −5.52439032265373369057358531895, −5.16260465009843706075090520212, −4.64775021617587382419607028476, −4.58946055911775104515313612123, −4.34444498840724671519273008958, −4.21889531714895978034052843759, −3.96549667906809138790946506277, −3.30284310987548295690518640383, −3.22637996615206476005784467529, −2.95538074429969910944654789652, −2.86134946042847222791301606996, −2.64737401194790779933805844914, −2.62445552557371477156189576065, −2.34122756250011474824872651977, −1.22085570205281891918273391839, −0.878399618539942468101899467964, −0.03696649653176492871749264361, 0.03696649653176492871749264361, 0.878399618539942468101899467964, 1.22085570205281891918273391839, 2.34122756250011474824872651977, 2.62445552557371477156189576065, 2.64737401194790779933805844914, 2.86134946042847222791301606996, 2.95538074429969910944654789652, 3.22637996615206476005784467529, 3.30284310987548295690518640383, 3.96549667906809138790946506277, 4.21889531714895978034052843759, 4.34444498840724671519273008958, 4.58946055911775104515313612123, 4.64775021617587382419607028476, 5.16260465009843706075090520212, 5.52439032265373369057358531895, 5.54925073888602436600916379674, 5.56109817647016539174293206730, 5.81889782441868448295012242101, 5.93387226490718014826913399535, 6.55365354972886238446239958690, 6.77670778135530802852219123197, 6.85915770406683865314038358087, 7.07794961465135876311224205509
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.