Properties

Label 24-384e12-1.1-c3e12-0-3
Degree $24$
Conductor $1.028\times 10^{31}$
Sign $1$
Analytic cond. $1.82964\times 10^{16}$
Root an. cond. $4.75990$
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  + 10·3-s + 50·9-s + 36·11-s + 120·23-s + 600·25-s + 78·27-s + 360·33-s − 528·37-s + 1.24e3·47-s + 1.58e3·49-s + 2.50e3·59-s − 624·61-s + 1.20e3·69-s + 2.04e3·71-s − 216·73-s + 6.00e3·75-s − 739·81-s + 4.57e3·83-s − 48·97-s + 1.80e3·99-s + 6.49e3·107-s − 144·109-s − 5.28e3·111-s − 6.69e3·121-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.92·3-s + 1.85·9-s + 0.986·11-s + 1.08·23-s + 24/5·25-s + 0.555·27-s + 1.89·33-s − 2.34·37-s + 3.87·47-s + 4.61·49-s + 5.53·59-s − 1.30·61-s + 2.09·69-s + 3.40·71-s − 0.346·73-s + 9.23·75-s − 1.01·81-s + 6.04·83-s − 0.0502·97-s + 1.82·99-s + 5.86·107-s − 0.126·109-s − 4.51·111-s − 5.03·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{84} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(1.82964\times 10^{16}\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{84} \cdot 3^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(351.8762958\)
\(L(\frac12)\) \(\approx\) \(351.8762958\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 10 T + 50 T^{2} - 26 p T^{3} - 67 p T^{4} + 124 p^{3} T^{5} - 1924 p^{2} T^{6} + 124 p^{6} T^{7} - 67 p^{7} T^{8} - 26 p^{10} T^{9} + 50 p^{12} T^{10} - 10 p^{15} T^{11} + p^{18} T^{12} \)
good5 \( 1 - 24 p^{2} T^{2} + 36642 p T^{4} - 33731768 T^{6} + 3958679103 T^{8} - 292014000048 T^{10} + 22017716382796 T^{12} - 292014000048 p^{6} T^{14} + 3958679103 p^{12} T^{16} - 33731768 p^{18} T^{18} + 36642 p^{25} T^{20} - 24 p^{32} T^{22} + p^{36} T^{24} \)
7 \( 1 - 1584 T^{2} + 1166442 T^{4} - 478563952 T^{6} + 86370897327 T^{8} + 18087151507872 T^{10} - 14510145747466484 T^{12} + 18087151507872 p^{6} T^{14} + 86370897327 p^{12} T^{16} - 478563952 p^{18} T^{18} + 1166442 p^{24} T^{20} - 1584 p^{30} T^{22} + p^{36} T^{24} \)
11 \( ( 1 - 18 T + 3834 T^{2} - 13302 T^{3} + 619797 p T^{4} + 39315780 T^{5} + 9651152012 T^{6} + 39315780 p^{3} T^{7} + 619797 p^{7} T^{8} - 13302 p^{9} T^{9} + 3834 p^{12} T^{10} - 18 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
13 \( ( 1 + 5730 T^{2} - 3072 p T^{3} + 19329879 T^{4} - 32093184 T^{5} + 3911101356 p T^{6} - 32093184 p^{3} T^{7} + 19329879 p^{6} T^{8} - 3072 p^{10} T^{9} + 5730 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
17 \( 1 - 31164 T^{2} + 484190562 T^{4} - 4991368612172 T^{6} + 38572842772586415 T^{8} - \)\(24\!\cdots\!92\)\( T^{10} + \)\(12\!\cdots\!56\)\( T^{12} - \)\(24\!\cdots\!92\)\( p^{6} T^{14} + 38572842772586415 p^{12} T^{16} - 4991368612172 p^{18} T^{18} + 484190562 p^{24} T^{20} - 31164 p^{30} T^{22} + p^{36} T^{24} \)
19 \( 1 - 38160 T^{2} + 808508346 T^{4} - 11916467660880 T^{6} + 135299522971061151 T^{8} - \)\(12\!\cdots\!08\)\( T^{10} + \)\(93\!\cdots\!44\)\( T^{12} - \)\(12\!\cdots\!08\)\( p^{6} T^{14} + 135299522971061151 p^{12} T^{16} - 11916467660880 p^{18} T^{18} + 808508346 p^{24} T^{20} - 38160 p^{30} T^{22} + p^{36} T^{24} \)
23 \( ( 1 - 60 T + 35946 T^{2} - 1305844 T^{3} + 618033375 T^{4} - 10828626072 T^{5} + 7737521174604 T^{6} - 10828626072 p^{3} T^{7} + 618033375 p^{6} T^{8} - 1305844 p^{9} T^{9} + 35946 p^{12} T^{10} - 60 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
29 \( 1 - 173784 T^{2} + 15604233546 T^{4} - 932142550689464 T^{6} + 41087275229333246751 T^{8} - \)\(14\!\cdots\!16\)\( T^{10} + \)\(38\!\cdots\!20\)\( T^{12} - \)\(14\!\cdots\!16\)\( p^{6} T^{14} + 41087275229333246751 p^{12} T^{16} - 932142550689464 p^{18} T^{18} + 15604233546 p^{24} T^{20} - 173784 p^{30} T^{22} + p^{36} T^{24} \)
31 \( 1 - 188448 T^{2} + 548969046 p T^{4} - 999868768809888 T^{6} + 44197970839051250895 T^{8} - \)\(16\!\cdots\!56\)\( T^{10} + \)\(51\!\cdots\!24\)\( T^{12} - \)\(16\!\cdots\!56\)\( p^{6} T^{14} + 44197970839051250895 p^{12} T^{16} - 999868768809888 p^{18} T^{18} + 548969046 p^{25} T^{20} - 188448 p^{30} T^{22} + p^{36} T^{24} \)
37 \( ( 1 + 264 T + 179778 T^{2} + 37541704 T^{3} + 13332243207 T^{4} + 2487709265808 T^{5} + 688884666730300 T^{6} + 2487709265808 p^{3} T^{7} + 13332243207 p^{6} T^{8} + 37541704 p^{9} T^{9} + 179778 p^{12} T^{10} + 264 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
41 \( 1 - 472860 T^{2} + 113880883458 T^{4} - 18423203395045100 T^{6} + \)\(22\!\cdots\!39\)\( T^{8} - \)\(21\!\cdots\!48\)\( T^{10} + \)\(16\!\cdots\!96\)\( T^{12} - \)\(21\!\cdots\!48\)\( p^{6} T^{14} + \)\(22\!\cdots\!39\)\( p^{12} T^{16} - 18423203395045100 p^{18} T^{18} + 113880883458 p^{24} T^{20} - 472860 p^{30} T^{22} + p^{36} T^{24} \)
43 \( 1 - 386256 T^{2} + 75858413850 T^{4} - 10754684841540112 T^{6} + \)\(12\!\cdots\!07\)\( T^{8} - \)\(12\!\cdots\!08\)\( T^{10} + \)\(10\!\cdots\!20\)\( T^{12} - \)\(12\!\cdots\!08\)\( p^{6} T^{14} + \)\(12\!\cdots\!07\)\( p^{12} T^{16} - 10754684841540112 p^{18} T^{18} + 75858413850 p^{24} T^{20} - 386256 p^{30} T^{22} + p^{36} T^{24} \)
47 \( ( 1 - 624 T + 9222 p T^{2} - 110814928 T^{3} + 31731910959 T^{4} + 3786299059104 T^{5} - 661034171971540 T^{6} + 3786299059104 p^{3} T^{7} + 31731910959 p^{6} T^{8} - 110814928 p^{9} T^{9} + 9222 p^{13} T^{10} - 624 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
53 \( 1 - 776088 T^{2} + 345553114986 T^{4} - 110076343120290296 T^{6} + \)\(27\!\cdots\!63\)\( T^{8} - \)\(54\!\cdots\!64\)\( T^{10} + \)\(88\!\cdots\!64\)\( T^{12} - \)\(54\!\cdots\!64\)\( p^{6} T^{14} + \)\(27\!\cdots\!63\)\( p^{12} T^{16} - 110076343120290296 p^{18} T^{18} + 345553114986 p^{24} T^{20} - 776088 p^{30} T^{22} + p^{36} T^{24} \)
59 \( ( 1 - 1254 T + 1118754 T^{2} - 560306738 T^{3} + 186995225991 T^{4} - 14913042447924 T^{5} - 4616030684849220 T^{6} - 14913042447924 p^{3} T^{7} + 186995225991 p^{6} T^{8} - 560306738 p^{9} T^{9} + 1118754 p^{12} T^{10} - 1254 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
61 \( ( 1 + 312 T + 8346 p T^{2} + 257561528 T^{3} + 211636873719 T^{4} + 86029951892976 T^{5} + 57402008417819932 T^{6} + 86029951892976 p^{3} T^{7} + 211636873719 p^{6} T^{8} + 257561528 p^{9} T^{9} + 8346 p^{13} T^{10} + 312 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
67 \( 1 - 2286816 T^{2} + 2411853842778 T^{4} - 1566286389213019744 T^{6} + \)\(71\!\cdots\!43\)\( T^{8} - \)\(25\!\cdots\!84\)\( T^{10} + \)\(80\!\cdots\!64\)\( T^{12} - \)\(25\!\cdots\!84\)\( p^{6} T^{14} + \)\(71\!\cdots\!43\)\( p^{12} T^{16} - 1566286389213019744 p^{18} T^{18} + 2411853842778 p^{24} T^{20} - 2286816 p^{30} T^{22} + p^{36} T^{24} \)
71 \( ( 1 - 1020 T + 2040810 T^{2} - 1572918772 T^{3} + 1765475696511 T^{4} - 1048763672220696 T^{5} + 832227063915098828 T^{6} - 1048763672220696 p^{3} T^{7} + 1765475696511 p^{6} T^{8} - 1572918772 p^{9} T^{9} + 2040810 p^{12} T^{10} - 1020 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
73 \( ( 1 + 108 T + 839826 T^{2} + 159624796 T^{3} + 499086952191 T^{4} + 64393086787416 T^{5} + 241123492212399420 T^{6} + 64393086787416 p^{3} T^{7} + 499086952191 p^{6} T^{8} + 159624796 p^{9} T^{9} + 839826 p^{12} T^{10} + 108 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
79 \( 1 - 4787328 T^{2} + 10929285260394 T^{4} - 15723000113198264960 T^{6} + \)\(15\!\cdots\!59\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(66\!\cdots\!56\)\( T^{12} - \)\(11\!\cdots\!88\)\( p^{6} T^{14} + \)\(15\!\cdots\!59\)\( p^{12} T^{16} - 15723000113198264960 p^{18} T^{18} + 10929285260394 p^{24} T^{20} - 4787328 p^{30} T^{22} + p^{36} T^{24} \)
83 \( ( 1 - 2286 T + 4665594 T^{2} - 6371132122 T^{3} + 7586066763543 T^{4} - 7182264432963204 T^{5} + 5999536723098321228 T^{6} - 7182264432963204 p^{3} T^{7} + 7586066763543 p^{6} T^{8} - 6371132122 p^{9} T^{9} + 4665594 p^{12} T^{10} - 2286 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
89 \( 1 - 4768524 T^{2} + 12015644253474 T^{4} - 230672060265750012 p T^{6} + \)\(26\!\cdots\!15\)\( T^{8} - \)\(25\!\cdots\!96\)\( T^{10} + \)\(20\!\cdots\!60\)\( T^{12} - \)\(25\!\cdots\!96\)\( p^{6} T^{14} + \)\(26\!\cdots\!15\)\( p^{12} T^{16} - 230672060265750012 p^{19} T^{18} + 12015644253474 p^{24} T^{20} - 4768524 p^{30} T^{22} + p^{36} T^{24} \)
97 \( ( 1 + 24 T + 3053178 T^{2} + 1259266232 T^{3} + 4503411377103 T^{4} + 2753933866197552 T^{5} + 48632588768677228 p T^{6} + 2753933866197552 p^{3} T^{7} + 4503411377103 p^{6} T^{8} + 1259266232 p^{9} T^{9} + 3053178 p^{12} T^{10} + 24 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
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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

−3.18879923532995388802470459273, −3.14726578574453180998684620967, −3.03515570403110466808961616113, −3.03094360912311243809094972105, −3.01503481128310974519102358630, −2.86049504091982944573727736154, −2.65735805568877116536486973973, −2.37409127944431197633157198347, −2.23258448020362551157766563034, −2.23020627252845294457641141491, −2.22254294448809636063580860442, −2.09316603216440046756264184648, −1.95597374755511047768285280581, −1.92326852211149962244417737984, −1.82509745727643049711576726172, −1.45991542948520451410517914772, −1.32398102661071219618847517017, −1.03220364771568435943649299204, −0.923701578131763946526472866100, −0.860753625846657256601063607050, −0.75183739521972265520206074725, −0.73987446957493397004581414162, −0.61112346466220863338819025837, −0.55082894371342139991919662256, −0.31225409414659223742680683795, 0.31225409414659223742680683795, 0.55082894371342139991919662256, 0.61112346466220863338819025837, 0.73987446957493397004581414162, 0.75183739521972265520206074725, 0.860753625846657256601063607050, 0.923701578131763946526472866100, 1.03220364771568435943649299204, 1.32398102661071219618847517017, 1.45991542948520451410517914772, 1.82509745727643049711576726172, 1.92326852211149962244417737984, 1.95597374755511047768285280581, 2.09316603216440046756264184648, 2.22254294448809636063580860442, 2.23020627252845294457641141491, 2.23258448020362551157766563034, 2.37409127944431197633157198347, 2.65735805568877116536486973973, 2.86049504091982944573727736154, 3.01503481128310974519102358630, 3.03094360912311243809094972105, 3.03515570403110466808961616113, 3.14726578574453180998684620967, 3.18879923532995388802470459273

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

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