Properties

Label 12-21e12-1.1-c1e6-0-1
Degree $12$
Conductor $7.356\times 10^{15}$
Sign $1$
Analytic cond. $1906.75$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 15·4-s − 3·5-s + 18·8-s − 18·10-s − 6·11-s + 3·13-s + 3·16-s − 6·17-s + 3·19-s − 45·20-s − 36·22-s − 12·23-s + 15·25-s + 18·26-s + 9·27-s − 9·29-s − 6·31-s − 30·32-s − 36·34-s + 3·37-s + 18·38-s − 54·40-s + 3·43-s − 90·44-s − 72·46-s + 6·47-s + ⋯
L(s)  = 1  + 4.24·2-s + 15/2·4-s − 1.34·5-s + 6.36·8-s − 5.69·10-s − 1.80·11-s + 0.832·13-s + 3/4·16-s − 1.45·17-s + 0.688·19-s − 10.0·20-s − 7.67·22-s − 2.50·23-s + 3·25-s + 3.53·26-s + 1.73·27-s − 1.67·29-s − 1.07·31-s − 5.30·32-s − 6.17·34-s + 0.493·37-s + 2.91·38-s − 8.53·40-s + 0.457·43-s − 13.5·44-s − 10.6·46-s + 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1906.75\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.925937984\)
\(L(\frac12)\) \(\approx\) \(3.925937984\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - p^{2} T^{3} + p^{3} T^{6} \)
7 \( 1 \)
good2 \( ( 1 - 3 T + 3 p T^{2} - 9 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
5 \( 1 + 3 T - 6 T^{2} - 9 T^{3} + 69 T^{4} + 6 p T^{5} - 371 T^{6} + 6 p^{2} T^{7} + 69 p^{2} T^{8} - 9 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 6 T - 6 T^{2} - 18 T^{3} + 492 T^{4} + 852 T^{5} - 2873 T^{6} + 852 p T^{7} + 492 p^{2} T^{8} - 18 p^{3} T^{9} - 6 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 3 T + 3 T^{2} - 76 T^{3} + 45 T^{4} + 135 T^{5} + 3246 T^{6} + 135 p T^{7} + 45 p^{2} T^{8} - 76 p^{3} T^{9} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 6 T - 24 T^{2} - 54 T^{3} + 1338 T^{4} + 1914 T^{5} - 18929 T^{6} + 1914 p T^{7} + 1338 p^{2} T^{8} - 54 p^{3} T^{9} - 24 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T - 42 T^{2} + 41 T^{3} + 1341 T^{4} - 216 T^{5} - 29541 T^{6} - 216 p T^{7} + 1341 p^{2} T^{8} + 41 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 12 T + 48 T^{2} + 54 T^{3} + 420 T^{4} + 6060 T^{5} + 37591 T^{6} + 6060 p T^{7} + 420 p^{2} T^{8} + 54 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 9 T + 30 T^{2} + 81 T^{3} - 579 T^{4} - 9414 T^{5} - 59051 T^{6} - 9414 p T^{7} - 579 p^{2} T^{8} + 81 p^{3} T^{9} + 30 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 3 T + 15 T^{2} - 137 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 3 T - 24 T^{2} - 301 T^{3} + 171 T^{4} + 6552 T^{5} + 58893 T^{6} + 6552 p T^{7} + 171 p^{2} T^{8} - 301 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 114 T^{2} - 18 T^{3} + 8322 T^{4} + 1026 T^{5} - 394913 T^{6} + 1026 p T^{7} + 8322 p^{2} T^{8} - 18 p^{3} T^{9} - 114 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 5670 p T^{7} + 9063 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 - 3 T + 87 T^{2} - 333 T^{3} + 87 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 + 6 T - 114 T^{2} - 378 T^{3} + 10716 T^{4} + 17304 T^{5} - 587549 T^{6} + 17304 p T^{7} + 10716 p^{2} T^{8} - 378 p^{3} T^{9} - 114 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 + 3 T + 105 T^{2} + 405 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 6 T + 168 T^{2} - 713 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 + 12 T + 222 T^{2} + 1591 T^{3} + 222 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 - 9 T + 159 T^{2} - 1305 T^{3} + 159 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 21 T + 138 T^{2} - 769 T^{3} + 10953 T^{4} - 30402 T^{5} - 450903 T^{6} - 30402 p T^{7} + 10953 p^{2} T^{8} - 769 p^{3} T^{9} + 138 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 + 21 T + 357 T^{2} + 3499 T^{3} + 357 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 18 T + 30 T^{2} + 702 T^{3} + 8088 T^{4} - 126648 T^{5} + 719359 T^{6} - 126648 p T^{7} + 8088 p^{2} T^{8} + 702 p^{3} T^{9} + 30 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 12 T - 60 T^{2} - 198 T^{3} + 7584 T^{4} - 70800 T^{5} - 1684181 T^{6} - 70800 p T^{7} + 7584 p^{2} T^{8} - 198 p^{3} T^{9} - 60 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 2421 T^{4} + 11502 T^{5} + 340233 T^{6} + 11502 p T^{7} + 2421 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.65399834147712095373816727147, −5.62594004901320904889964360985, −5.52996336151396633458072319392, −5.34966852562544007477651273867, −5.34202822079052198010951961552, −4.89114962578282056372463793187, −4.85554584725551785747773197947, −4.80731246549708467716868329916, −4.52269838339875983186760250700, −4.24520980253784602059303601803, −4.12621263038470736962608168780, −4.12173679141026505828035303466, −4.10203968912438284950744404112, −3.68941037594969802968216591028, −3.51186819505681221662884877115, −3.38157216903191666872861020199, −3.09697582623663454091332160905, −2.95724197247819951691978530924, −2.61051943714259769912099645038, −2.52109541255825529462245258740, −2.26320601607568600418535310076, −1.68847135581890028304984931045, −1.66855144921805087931143060598, −0.882887523815293774766513605622, −0.24802698083798666870835540963, 0.24802698083798666870835540963, 0.882887523815293774766513605622, 1.66855144921805087931143060598, 1.68847135581890028304984931045, 2.26320601607568600418535310076, 2.52109541255825529462245258740, 2.61051943714259769912099645038, 2.95724197247819951691978530924, 3.09697582623663454091332160905, 3.38157216903191666872861020199, 3.51186819505681221662884877115, 3.68941037594969802968216591028, 4.10203968912438284950744404112, 4.12173679141026505828035303466, 4.12621263038470736962608168780, 4.24520980253784602059303601803, 4.52269838339875983186760250700, 4.80731246549708467716868329916, 4.85554584725551785747773197947, 4.89114962578282056372463793187, 5.34202822079052198010951961552, 5.34966852562544007477651273867, 5.52996336151396633458072319392, 5.62594004901320904889964360985, 5.65399834147712095373816727147

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

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