Properties

Label 12-304e6-1.1-c9e6-0-1
Degree $12$
Conductor $7.893\times 10^{14}$
Sign $1$
Analytic cond. $1.47321\times 10^{13}$
Root an. cond. $12.5128$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 155·3-s − 3.61e3·5-s − 4.08e3·7-s − 3.82e4·9-s + 6.93e4·11-s − 1.91e5·13-s − 5.59e5·15-s − 2.88e5·17-s + 7.81e5·19-s − 6.33e5·21-s + 5.06e4·23-s + 4.22e6·25-s − 9.46e6·27-s − 7.17e6·29-s + 4.31e6·31-s + 1.07e7·33-s + 1.47e7·35-s − 1.56e7·37-s − 2.97e7·39-s − 2.51e7·41-s + 7.51e7·43-s + 1.38e8·45-s + 8.37e7·47-s − 1.02e8·49-s − 4.46e7·51-s − 1.38e8·53-s − 2.50e8·55-s + ⋯
L(s)  = 1  + 1.10·3-s − 2.58·5-s − 0.643·7-s − 1.94·9-s + 1.42·11-s − 1.86·13-s − 2.85·15-s − 0.836·17-s + 1.37·19-s − 0.710·21-s + 0.0377·23-s + 2.16·25-s − 3.42·27-s − 1.88·29-s + 0.838·31-s + 1.57·33-s + 1.66·35-s − 1.37·37-s − 2.05·39-s − 1.38·41-s + 3.35·43-s + 5.01·45-s + 2.50·47-s − 2.53·49-s − 0.924·51-s − 2.40·53-s − 3.68·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(1.47321\times 10^{13}\)
Root analytic conductor: \(12.5128\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{24} \cdot 19^{6} ,\ ( \ : [9/2]^{6} ),\ 1 )\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( ( 1 - p^{4} T )^{6} \)
good3 \( 1 - 155 T + 62249 T^{2} - 6105562 T^{3} + 636046237 p T^{4} - 19386269207 p^{2} T^{5} + 1680430278122 p^{3} T^{6} - 19386269207 p^{11} T^{7} + 636046237 p^{19} T^{8} - 6105562 p^{27} T^{9} + 62249 p^{36} T^{10} - 155 p^{45} T^{11} + p^{54} T^{12} \)
5 \( 1 + 3612 T + 8819203 T^{2} + 15226118454 T^{3} + 4491105434991 p T^{4} + 1174269510249486 p^{2} T^{5} + 335172475640149498 p^{3} T^{6} + 1174269510249486 p^{11} T^{7} + 4491105434991 p^{19} T^{8} + 15226118454 p^{27} T^{9} + 8819203 p^{36} T^{10} + 3612 p^{45} T^{11} + p^{54} T^{12} \)
7 \( 1 + 4085 T + 16985632 p T^{2} + 12594232429 p^{2} T^{3} + 20742144301696 p^{3} T^{4} + 16781313847414901 p^{4} T^{5} + 18718742091583832402 p^{5} T^{6} + 16781313847414901 p^{13} T^{7} + 20742144301696 p^{21} T^{8} + 12594232429 p^{29} T^{9} + 16985632 p^{37} T^{10} + 4085 p^{45} T^{11} + p^{54} T^{12} \)
11 \( 1 - 69312 T + 8919448955 T^{2} - 362406965864970 T^{3} + 30850521292803957655 T^{4} - \)\(90\!\cdots\!82\)\( T^{5} + \)\(76\!\cdots\!30\)\( T^{6} - \)\(90\!\cdots\!82\)\( p^{9} T^{7} + 30850521292803957655 p^{18} T^{8} - 362406965864970 p^{27} T^{9} + 8919448955 p^{36} T^{10} - 69312 p^{45} T^{11} + p^{54} T^{12} \)
13 \( 1 + 191747 T + 63256423831 T^{2} + 8493310736420104 T^{3} + \)\(16\!\cdots\!05\)\( T^{4} + \)\(16\!\cdots\!69\)\( T^{5} + \)\(22\!\cdots\!62\)\( T^{6} + \)\(16\!\cdots\!69\)\( p^{9} T^{7} + \)\(16\!\cdots\!05\)\( p^{18} T^{8} + 8493310736420104 p^{27} T^{9} + 63256423831 p^{36} T^{10} + 191747 p^{45} T^{11} + p^{54} T^{12} \)
17 \( 1 + 288195 T + 584693721722 T^{2} + 128890679515643145 T^{3} + \)\(88\!\cdots\!76\)\( p T^{4} + \)\(26\!\cdots\!07\)\( T^{5} + \)\(22\!\cdots\!20\)\( T^{6} + \)\(26\!\cdots\!07\)\( p^{9} T^{7} + \)\(88\!\cdots\!76\)\( p^{19} T^{8} + 128890679515643145 p^{27} T^{9} + 584693721722 p^{36} T^{10} + 288195 p^{45} T^{11} + p^{54} T^{12} \)
23 \( 1 - 50697 T + 7848341516961 T^{2} - 2213517228317023194 T^{3} + \)\(28\!\cdots\!23\)\( T^{4} - \)\(10\!\cdots\!73\)\( T^{5} + \)\(61\!\cdots\!46\)\( T^{6} - \)\(10\!\cdots\!73\)\( p^{9} T^{7} + \)\(28\!\cdots\!23\)\( p^{18} T^{8} - 2213517228317023194 p^{27} T^{9} + 7848341516961 p^{36} T^{10} - 50697 p^{45} T^{11} + p^{54} T^{12} \)
29 \( 1 + 7178667 T + 44233248186399 T^{2} + \)\(16\!\cdots\!68\)\( T^{3} + \)\(62\!\cdots\!81\)\( T^{4} + \)\(12\!\cdots\!89\)\( T^{5} + \)\(43\!\cdots\!58\)\( T^{6} + \)\(12\!\cdots\!89\)\( p^{9} T^{7} + \)\(62\!\cdots\!81\)\( p^{18} T^{8} + \)\(16\!\cdots\!68\)\( p^{27} T^{9} + 44233248186399 p^{36} T^{10} + 7178667 p^{45} T^{11} + p^{54} T^{12} \)
31 \( 1 - 4310716 T + 91495602933162 T^{2} - \)\(39\!\cdots\!76\)\( T^{3} + \)\(48\!\cdots\!55\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{5} + \)\(15\!\cdots\!00\)\( T^{6} - \)\(17\!\cdots\!20\)\( p^{9} T^{7} + \)\(48\!\cdots\!55\)\( p^{18} T^{8} - \)\(39\!\cdots\!76\)\( p^{27} T^{9} + 91495602933162 p^{36} T^{10} - 4310716 p^{45} T^{11} + p^{54} T^{12} \)
37 \( 1 + 15687040 T + 477200595260658 T^{2} + \)\(41\!\cdots\!12\)\( T^{3} + \)\(77\!\cdots\!91\)\( T^{4} + \)\(42\!\cdots\!68\)\( T^{5} + \)\(89\!\cdots\!12\)\( T^{6} + \)\(42\!\cdots\!68\)\( p^{9} T^{7} + \)\(77\!\cdots\!91\)\( p^{18} T^{8} + \)\(41\!\cdots\!12\)\( p^{27} T^{9} + 477200595260658 p^{36} T^{10} + 15687040 p^{45} T^{11} + p^{54} T^{12} \)
41 \( 1 + 25134306 T + 1079097703143394 T^{2} + \)\(15\!\cdots\!50\)\( T^{3} + \)\(49\!\cdots\!67\)\( T^{4} + \)\(55\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!04\)\( T^{6} + \)\(55\!\cdots\!00\)\( p^{9} T^{7} + \)\(49\!\cdots\!67\)\( p^{18} T^{8} + \)\(15\!\cdots\!50\)\( p^{27} T^{9} + 1079097703143394 p^{36} T^{10} + 25134306 p^{45} T^{11} + p^{54} T^{12} \)
43 \( 1 - 75118674 T + 4425622451146347 T^{2} - \)\(16\!\cdots\!02\)\( T^{3} + \)\(56\!\cdots\!59\)\( T^{4} - \)\(14\!\cdots\!44\)\( T^{5} + \)\(36\!\cdots\!98\)\( T^{6} - \)\(14\!\cdots\!44\)\( p^{9} T^{7} + \)\(56\!\cdots\!59\)\( p^{18} T^{8} - \)\(16\!\cdots\!02\)\( p^{27} T^{9} + 4425622451146347 p^{36} T^{10} - 75118674 p^{45} T^{11} + p^{54} T^{12} \)
47 \( 1 - 83731938 T + 7369210577066907 T^{2} - \)\(39\!\cdots\!30\)\( T^{3} + \)\(21\!\cdots\!47\)\( T^{4} - \)\(82\!\cdots\!24\)\( T^{5} + \)\(31\!\cdots\!90\)\( T^{6} - \)\(82\!\cdots\!24\)\( p^{9} T^{7} + \)\(21\!\cdots\!47\)\( p^{18} T^{8} - \)\(39\!\cdots\!30\)\( p^{27} T^{9} + 7369210577066907 p^{36} T^{10} - 83731938 p^{45} T^{11} + p^{54} T^{12} \)
53 \( 1 + 138019203 T + 23264355593581535 T^{2} + \)\(20\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!13\)\( T^{4} + \)\(13\!\cdots\!53\)\( T^{5} + \)\(89\!\cdots\!46\)\( T^{6} + \)\(13\!\cdots\!53\)\( p^{9} T^{7} + \)\(20\!\cdots\!13\)\( p^{18} T^{8} + \)\(20\!\cdots\!20\)\( p^{27} T^{9} + 23264355593581535 p^{36} T^{10} + 138019203 p^{45} T^{11} + p^{54} T^{12} \)
59 \( 1 - 7809915 T + 42493160832721529 T^{2} - \)\(13\!\cdots\!18\)\( T^{3} + \)\(81\!\cdots\!11\)\( T^{4} - \)\(10\!\cdots\!47\)\( T^{5} + \)\(90\!\cdots\!38\)\( T^{6} - \)\(10\!\cdots\!47\)\( p^{9} T^{7} + \)\(81\!\cdots\!11\)\( p^{18} T^{8} - \)\(13\!\cdots\!18\)\( p^{27} T^{9} + 42493160832721529 p^{36} T^{10} - 7809915 p^{45} T^{11} + p^{54} T^{12} \)
61 \( 1 - 191946566 T + 54780670024444367 T^{2} - \)\(82\!\cdots\!90\)\( T^{3} + \)\(13\!\cdots\!87\)\( T^{4} - \)\(16\!\cdots\!64\)\( T^{5} + \)\(20\!\cdots\!70\)\( T^{6} - \)\(16\!\cdots\!64\)\( p^{9} T^{7} + \)\(13\!\cdots\!87\)\( p^{18} T^{8} - \)\(82\!\cdots\!90\)\( p^{27} T^{9} + 54780670024444367 p^{36} T^{10} - 191946566 p^{45} T^{11} + p^{54} T^{12} \)
67 \( 1 - 16109787 T + 108574986848183601 T^{2} - \)\(72\!\cdots\!50\)\( T^{3} + \)\(54\!\cdots\!43\)\( T^{4} - \)\(92\!\cdots\!51\)\( T^{5} + \)\(17\!\cdots\!14\)\( T^{6} - \)\(92\!\cdots\!51\)\( p^{9} T^{7} + \)\(54\!\cdots\!43\)\( p^{18} T^{8} - \)\(72\!\cdots\!50\)\( p^{27} T^{9} + 108574986848183601 p^{36} T^{10} - 16109787 p^{45} T^{11} + p^{54} T^{12} \)
71 \( 1 - 264469698 T + 176605075037501922 T^{2} - \)\(32\!\cdots\!74\)\( T^{3} + \)\(11\!\cdots\!55\)\( T^{4} - \)\(16\!\cdots\!12\)\( T^{5} + \)\(47\!\cdots\!36\)\( T^{6} - \)\(16\!\cdots\!12\)\( p^{9} T^{7} + \)\(11\!\cdots\!55\)\( p^{18} T^{8} - \)\(32\!\cdots\!74\)\( p^{27} T^{9} + 176605075037501922 p^{36} T^{10} - 264469698 p^{45} T^{11} + p^{54} T^{12} \)
73 \( 1 + 287572857 T + 189879359499479490 T^{2} + \)\(45\!\cdots\!91\)\( T^{3} + \)\(19\!\cdots\!20\)\( T^{4} + \)\(39\!\cdots\!49\)\( T^{5} + \)\(12\!\cdots\!48\)\( T^{6} + \)\(39\!\cdots\!49\)\( p^{9} T^{7} + \)\(19\!\cdots\!20\)\( p^{18} T^{8} + \)\(45\!\cdots\!91\)\( p^{27} T^{9} + 189879359499479490 p^{36} T^{10} + 287572857 p^{45} T^{11} + p^{54} T^{12} \)
79 \( 1 - 86534002 T + 338621828328018454 T^{2} - \)\(30\!\cdots\!94\)\( T^{3} + \)\(40\!\cdots\!31\)\( T^{4} + \)\(50\!\cdots\!48\)\( T^{5} + \)\(35\!\cdots\!28\)\( T^{6} + \)\(50\!\cdots\!48\)\( p^{9} T^{7} + \)\(40\!\cdots\!31\)\( p^{18} T^{8} - \)\(30\!\cdots\!94\)\( p^{27} T^{9} + 338621828328018454 p^{36} T^{10} - 86534002 p^{45} T^{11} + p^{54} T^{12} \)
83 \( 1 - 359214570 T + 415846672018866310 T^{2} - \)\(17\!\cdots\!18\)\( T^{3} + \)\(16\!\cdots\!51\)\( T^{4} - \)\(52\!\cdots\!56\)\( T^{5} + \)\(32\!\cdots\!32\)\( T^{6} - \)\(52\!\cdots\!56\)\( p^{9} T^{7} + \)\(16\!\cdots\!51\)\( p^{18} T^{8} - \)\(17\!\cdots\!18\)\( p^{27} T^{9} + 415846672018866310 p^{36} T^{10} - 359214570 p^{45} T^{11} + p^{54} T^{12} \)
89 \( 1 + 2263866306 T + 3767251604053176142 T^{2} + \)\(43\!\cdots\!58\)\( T^{3} + \)\(41\!\cdots\!19\)\( T^{4} + \)\(31\!\cdots\!04\)\( T^{5} + \)\(20\!\cdots\!16\)\( T^{6} + \)\(31\!\cdots\!04\)\( p^{9} T^{7} + \)\(41\!\cdots\!19\)\( p^{18} T^{8} + \)\(43\!\cdots\!58\)\( p^{27} T^{9} + 3767251604053176142 p^{36} T^{10} + 2263866306 p^{45} T^{11} + p^{54} T^{12} \)
97 \( 1 - 2705893460 T + 5799960519982386998 T^{2} - \)\(86\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!43\)\( T^{4} - \)\(12\!\cdots\!48\)\( T^{5} + \)\(11\!\cdots\!52\)\( T^{6} - \)\(12\!\cdots\!48\)\( p^{9} T^{7} + \)\(11\!\cdots\!43\)\( p^{18} T^{8} - \)\(86\!\cdots\!36\)\( p^{27} T^{9} + 5799960519982386998 p^{36} T^{10} - 2705893460 p^{45} T^{11} + p^{54} T^{12} \)
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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

−5.32606015785312617099435563727, −4.97439979371339355762440700549, −4.90743031563687052132025268645, −4.76819836527642734393239583949, −4.69831489565144513795432449396, −4.28017901591967880003067367610, −4.09826571087795917041491060977, −3.99363191227807002934032800982, −3.76195071122511102693228578783, −3.57105788946479838788758252205, −3.52169047662600886328513999728, −3.47366415324261029659972736265, −3.28790831304999798092216640025, −2.84560355705821983463752193140, −2.80836092845989985703871622713, −2.48168396537269024744054215630, −2.37521642072884107727010256541, −2.34198832648923824751133753948, −2.29274393113782652630247831485, −1.73130648479387067343484345335, −1.40539078003259051662459946193, −1.36189063721696545706570388715, −1.14173151201810702730655083668, −0.866866338636338954942571721172, −0.74538495576022260711615726037, 0, 0, 0, 0, 0, 0, 0.74538495576022260711615726037, 0.866866338636338954942571721172, 1.14173151201810702730655083668, 1.36189063721696545706570388715, 1.40539078003259051662459946193, 1.73130648479387067343484345335, 2.29274393113782652630247831485, 2.34198832648923824751133753948, 2.37521642072884107727010256541, 2.48168396537269024744054215630, 2.80836092845989985703871622713, 2.84560355705821983463752193140, 3.28790831304999798092216640025, 3.47366415324261029659972736265, 3.52169047662600886328513999728, 3.57105788946479838788758252205, 3.76195071122511102693228578783, 3.99363191227807002934032800982, 4.09826571087795917041491060977, 4.28017901591967880003067367610, 4.69831489565144513795432449396, 4.76819836527642734393239583949, 4.90743031563687052132025268645, 4.97439979371339355762440700549, 5.32606015785312617099435563727

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

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