Properties

Label 24-21e12-1.1-c3e12-0-0
Degree $24$
Conductor $7.356\times 10^{15}$
Sign $1$
Analytic cond. $13.0925$
Root an. cond. $1.11312$
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  − 3·3-s − 17·4-s − 56·7-s + 3·9-s + 51·12-s + 206·16-s + 300·19-s + 168·21-s + 354·25-s + 952·28-s − 930·31-s − 51·36-s + 764·37-s − 1.01e3·43-s − 618·48-s + 1.40e3·49-s − 900·57-s + 2.35e3·61-s − 168·63-s − 2.31e3·64-s + 792·67-s − 2.90e3·73-s − 1.06e3·75-s − 5.10e3·76-s + 1.67e3·79-s + 207·81-s − 2.85e3·84-s + ⋯
L(s)  = 1  − 0.577·3-s − 2.12·4-s − 3.02·7-s + 1/9·9-s + 1.22·12-s + 3.21·16-s + 3.62·19-s + 1.74·21-s + 2.83·25-s + 6.42·28-s − 5.38·31-s − 0.236·36-s + 3.39·37-s − 3.58·43-s − 1.85·48-s + 4.08·49-s − 2.09·57-s + 4.94·61-s − 0.335·63-s − 4.51·64-s + 1.44·67-s − 4.65·73-s − 1.63·75-s − 7.69·76-s + 2.38·79-s + 0.283·81-s − 3.70·84-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(13.0925\)
Root analytic conductor: \(1.11312\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 7^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.3918469809\)
\(L(\frac12)\) \(\approx\) \(0.3918469809\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p T + 2 p T^{2} + p^{2} T^{3} - 22 p^{2} T^{4} - 95 p^{3} T^{5} - 1334 p^{3} T^{6} - 95 p^{6} T^{7} - 22 p^{8} T^{8} + p^{11} T^{9} + 2 p^{13} T^{10} + p^{16} T^{11} + p^{18} T^{12} \)
7 \( ( 1 + 4 p T + 68 p T^{2} + 220 p^{2} T^{3} + 68 p^{4} T^{4} + 4 p^{7} T^{5} + p^{9} T^{6} )^{2} \)
good2 \( 1 + 17 T^{2} + 83 T^{4} + 111 p T^{6} + 343 p^{2} T^{8} - 2549 p^{4} T^{10} - 10715 p^{6} T^{12} - 2549 p^{10} T^{14} + 343 p^{14} T^{16} + 111 p^{19} T^{18} + 83 p^{24} T^{20} + 17 p^{30} T^{22} + p^{36} T^{24} \)
5 \( 1 - 354 T^{2} + 53346 T^{4} - 3913744 T^{6} + 238386078 T^{8} - 13245019542 p T^{10} + 12349753291374 T^{12} - 13245019542 p^{7} T^{14} + 238386078 p^{12} T^{16} - 3913744 p^{18} T^{18} + 53346 p^{24} T^{20} - 354 p^{30} T^{22} + p^{36} T^{24} \)
11 \( 1 + 4742 T^{2} + 10147682 T^{4} + 18336975648 T^{6} + 35136366245806 T^{8} + 55422032379705850 T^{10} + 74318354148055569358 T^{12} + 55422032379705850 p^{6} T^{14} + 35136366245806 p^{12} T^{16} + 18336975648 p^{18} T^{18} + 10147682 p^{24} T^{20} + 4742 p^{30} T^{22} + p^{36} T^{24} \)
13 \( ( 1 - 8847 T^{2} + 36037359 T^{4} - 94069474274 T^{6} + 36037359 p^{6} T^{8} - 8847 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
17 \( 1 - 17217 T^{2} + 160729722 T^{4} - 865949822635 T^{6} + 2480649769543932 T^{8} + 2994987661331912811 T^{10} - \)\(48\!\cdots\!48\)\( T^{12} + 2994987661331912811 p^{6} T^{14} + 2480649769543932 p^{12} T^{16} - 865949822635 p^{18} T^{18} + 160729722 p^{24} T^{20} - 17217 p^{30} T^{22} + p^{36} T^{24} \)
19 \( ( 1 - 150 T + 30330 T^{2} - 3424500 T^{3} + 468855630 T^{4} - 40538051670 T^{5} + 4025485422214 T^{6} - 40538051670 p^{3} T^{7} + 468855630 p^{6} T^{8} - 3424500 p^{9} T^{9} + 30330 p^{12} T^{10} - 150 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
23 \( 1 + 58691 T^{2} + 1855192226 T^{4} + 43669490183457 T^{6} + 835010896549582396 T^{8} + \)\(13\!\cdots\!91\)\( T^{10} + \)\(17\!\cdots\!88\)\( T^{12} + \)\(13\!\cdots\!91\)\( p^{6} T^{14} + 835010896549582396 p^{12} T^{16} + 43669490183457 p^{18} T^{18} + 1855192226 p^{24} T^{20} + 58691 p^{30} T^{22} + p^{36} T^{24} \)
29 \( ( 1 - 26333 T^{2} + 912806819 T^{4} - 27528483699758 T^{6} + 912806819 p^{6} T^{8} - 26333 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
31 \( ( 1 + 15 p T + 176469 T^{2} + 1565910 p T^{3} + 12033403527 T^{4} + 2446560037821 T^{5} + 457872227799046 T^{6} + 2446560037821 p^{3} T^{7} + 12033403527 p^{6} T^{8} + 1565910 p^{10} T^{9} + 176469 p^{12} T^{10} + 15 p^{16} T^{11} + p^{18} T^{12} )^{2} \)
37 \( ( 1 - 382 T - 886 p T^{2} + 5434056 T^{3} + 8807187550 T^{4} - 649676877530 T^{5} - 374210188787594 T^{6} - 649676877530 p^{3} T^{7} + 8807187550 p^{6} T^{8} + 5434056 p^{9} T^{9} - 886 p^{13} T^{10} - 382 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
41 \( ( 1 + 240738 T^{2} + 28558186959 T^{4} + 1368012595132 p^{2} T^{6} + 28558186959 p^{6} T^{8} + 240738 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
43 \( ( 1 + 253 T + 215237 T^{2} + 33567598 T^{3} + 215237 p^{3} T^{4} + 253 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
47 \( 1 - 437385 T^{2} + 98505882894 T^{4} - 16642251751296271 T^{6} + \)\(23\!\cdots\!92\)\( T^{8} - \)\(28\!\cdots\!97\)\( T^{10} + \)\(30\!\cdots\!08\)\( T^{12} - \)\(28\!\cdots\!97\)\( p^{6} T^{14} + \)\(23\!\cdots\!92\)\( p^{12} T^{16} - 16642251751296271 p^{18} T^{18} + 98505882894 p^{24} T^{20} - 437385 p^{30} T^{22} + p^{36} T^{24} \)
53 \( 1 + 362162 T^{2} + 31342470242 T^{4} + 2991238457561520 T^{6} + \)\(17\!\cdots\!22\)\( T^{8} + \)\(23\!\cdots\!54\)\( T^{10} + \)\(11\!\cdots\!62\)\( T^{12} + \)\(23\!\cdots\!54\)\( p^{6} T^{14} + \)\(17\!\cdots\!22\)\( p^{12} T^{16} + 2991238457561520 p^{18} T^{18} + 31342470242 p^{24} T^{20} + 362162 p^{30} T^{22} + p^{36} T^{24} \)
59 \( 1 - 661854 T^{2} + 202275596298 T^{4} - 40619516236130848 T^{6} + \)\(74\!\cdots\!38\)\( T^{8} - \)\(16\!\cdots\!86\)\( T^{10} + \)\(35\!\cdots\!06\)\( T^{12} - \)\(16\!\cdots\!86\)\( p^{6} T^{14} + \)\(74\!\cdots\!38\)\( p^{12} T^{16} - 40619516236130848 p^{18} T^{18} + 202275596298 p^{24} T^{20} - 661854 p^{30} T^{22} + p^{36} T^{24} \)
61 \( ( 1 - 1179 T + 941982 T^{2} - 564310665 T^{3} + 276007099296 T^{4} - 128710086478551 T^{5} + 56963412393227764 T^{6} - 128710086478551 p^{3} T^{7} + 276007099296 p^{6} T^{8} - 564310665 p^{9} T^{9} + 941982 p^{12} T^{10} - 1179 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
67 \( ( 1 - 396 T - 208818 T^{2} + 528240452 T^{3} - 109541196954 T^{4} - 71767532435832 T^{5} + 113262131816538126 T^{6} - 71767532435832 p^{3} T^{7} - 109541196954 p^{6} T^{8} + 528240452 p^{9} T^{9} - 208818 p^{12} T^{10} - 396 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
71 \( ( 1 - 1922318 T^{2} + 1602890240687 T^{4} - 746575415906526884 T^{6} + 1602890240687 p^{6} T^{8} - 1922318 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
73 \( ( 1 + 1452 T + 1911888 T^{2} + 1755642240 T^{3} + 1485047372688 T^{4} + 1012976615051412 T^{5} + 691104732915920110 T^{6} + 1012976615051412 p^{3} T^{7} + 1485047372688 p^{6} T^{8} + 1755642240 p^{9} T^{9} + 1911888 p^{12} T^{10} + 1452 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
79 \( ( 1 - 837 T - 518715 T^{2} + 592038158 T^{3} + 210634653159 T^{4} - 180227265033825 T^{5} - 13471835399034906 T^{6} - 180227265033825 p^{3} T^{7} + 210634653159 p^{6} T^{8} + 592038158 p^{9} T^{9} - 518715 p^{12} T^{10} - 837 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
83 \( ( 1 + 2862735 T^{2} + 3680795850915 T^{4} + 2710866692924348218 T^{6} + 3680795850915 p^{6} T^{8} + 2862735 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
89 \( 1 - 1635561 T^{2} + 1787180462010 T^{4} - 817237014874197619 T^{6} - \)\(17\!\cdots\!36\)\( T^{8} + \)\(75\!\cdots\!95\)\( T^{10} - \)\(70\!\cdots\!84\)\( T^{12} + \)\(75\!\cdots\!95\)\( p^{6} T^{14} - \)\(17\!\cdots\!36\)\( p^{12} T^{16} - 817237014874197619 p^{18} T^{18} + 1787180462010 p^{24} T^{20} - 1635561 p^{30} T^{22} + p^{36} T^{24} \)
97 \( ( 1 - 3316347 T^{2} + 4922505333747 T^{4} - 4971002050523297522 T^{6} + 4922505333747 p^{6} T^{8} - 3316347 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.85122863260842460225129262524, −6.78596493212918899473536583037, −6.54424658556538739141496036566, −6.48798872909125731404408245587, −6.11556571519401314145195555042, −5.95681458648031482662014012012, −5.80657903662348091305374854425, −5.49556994590938450967968799317, −5.36456384693872534346677849291, −5.35183005861158463724254213436, −5.28992533284485955674918591454, −5.23094047433692741916386119015, −4.72408387646424694577468886017, −4.48343967413805275217402009652, −4.35196236005114589951852683636, −4.06378845584135650286032186020, −3.79934788836565004940873003757, −3.45078836605789536516177156316, −3.34255824695526558205209022964, −3.26619686304507911137097675648, −3.13568813027488427376538127440, −2.77228712040942353118855861849, −1.99943855337412098492483906890, −1.07429463353818485527786137358, −0.49993227456804802842988061035, 0.49993227456804802842988061035, 1.07429463353818485527786137358, 1.99943855337412098492483906890, 2.77228712040942353118855861849, 3.13568813027488427376538127440, 3.26619686304507911137097675648, 3.34255824695526558205209022964, 3.45078836605789536516177156316, 3.79934788836565004940873003757, 4.06378845584135650286032186020, 4.35196236005114589951852683636, 4.48343967413805275217402009652, 4.72408387646424694577468886017, 5.23094047433692741916386119015, 5.28992533284485955674918591454, 5.35183005861158463724254213436, 5.36456384693872534346677849291, 5.49556994590938450967968799317, 5.80657903662348091305374854425, 5.95681458648031482662014012012, 6.11556571519401314145195555042, 6.48798872909125731404408245587, 6.54424658556538739141496036566, 6.78596493212918899473536583037, 6.85122863260842460225129262524

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

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