Properties

Label 12-21e12-1.1-c3e6-0-0
Degree $12$
Conductor $7.356\times 10^{15}$
Sign $1$
Analytic cond. $3.10333\times 10^{8}$
Root an. cond. $5.10096$
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  + 2-s − 11·5-s − 21·8-s − 11·10-s + 35·11-s − 124·13-s − 31·16-s − 48·17-s − 202·19-s + 35·22-s + 216·23-s + 183·25-s − 124·26-s − 106·29-s − 95·31-s + 64·32-s − 48·34-s − 262·37-s − 202·38-s + 231·40-s + 488·41-s + 720·43-s + 216·46-s + 210·47-s + 183·50-s + 393·53-s − 385·55-s + ⋯
L(s)  = 1  + 0.353·2-s − 0.983·5-s − 0.928·8-s − 0.347·10-s + 0.959·11-s − 2.64·13-s − 0.484·16-s − 0.684·17-s − 2.43·19-s + 0.339·22-s + 1.95·23-s + 1.46·25-s − 0.935·26-s − 0.678·29-s − 0.550·31-s + 0.353·32-s − 0.242·34-s − 1.16·37-s − 0.862·38-s + 0.913·40-s + 1.85·41-s + 2.55·43-s + 0.692·46-s + 0.651·47-s + 0.517·50-s + 1.01·53-s − 0.943·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(3.10333\times 10^{8}\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.01184446372\)
\(L(\frac12)\) \(\approx\) \(0.01184446372\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - T + T^{2} + 5 p^{2} T^{3} - 5 p T^{4} - p^{6} T^{5} + 265 p^{2} T^{6} - p^{9} T^{7} - 5 p^{7} T^{8} + 5 p^{11} T^{9} + p^{12} T^{10} - p^{15} T^{11} + p^{18} T^{12} \)
5 \( 1 + 11 T - 62 T^{2} - 203 p T^{3} - 1208 p T^{4} - 54313 T^{5} + 121696 T^{6} - 54313 p^{3} T^{7} - 1208 p^{7} T^{8} - 203 p^{10} T^{9} - 62 p^{12} T^{10} + 11 p^{15} T^{11} + p^{18} T^{12} \)
11 \( 1 - 35 T - 1400 T^{2} + 113593 T^{3} - 198940 T^{4} - 87110135 T^{5} + 3928586038 T^{6} - 87110135 p^{3} T^{7} - 198940 p^{6} T^{8} + 113593 p^{9} T^{9} - 1400 p^{12} T^{10} - 35 p^{15} T^{11} + p^{18} T^{12} \)
13 \( ( 1 + 62 T + 7016 T^{2} + 253976 T^{3} + 7016 p^{3} T^{4} + 62 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
17 \( 1 + 48 T - 10035 T^{2} - 125232 T^{3} + 74409318 T^{4} - 234420432 T^{5} - 437742983351 T^{6} - 234420432 p^{3} T^{7} + 74409318 p^{6} T^{8} - 125232 p^{9} T^{9} - 10035 p^{12} T^{10} + 48 p^{15} T^{11} + p^{18} T^{12} \)
19 \( 1 + 202 T + 7946 T^{2} + 627636 T^{3} + 247297462 T^{4} + 17185599794 T^{5} + 349471935958 T^{6} + 17185599794 p^{3} T^{7} + 247297462 p^{6} T^{8} + 627636 p^{9} T^{9} + 7946 p^{12} T^{10} + 202 p^{15} T^{11} + p^{18} T^{12} \)
23 \( 1 - 216 T + 10827 T^{2} - 387864 T^{3} + 53856198 T^{4} + 24653558952 T^{5} - 5413409425505 T^{6} + 24653558952 p^{3} T^{7} + 53856198 p^{6} T^{8} - 387864 p^{9} T^{9} + 10827 p^{12} T^{10} - 216 p^{15} T^{11} + p^{18} T^{12} \)
29 \( ( 1 + 53 T + 52695 T^{2} + 3410210 T^{3} + 52695 p^{3} T^{4} + 53 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
31 \( 1 + 95 T - 70347 T^{2} - 3756594 T^{3} + 3398738767 T^{4} + 83374434539 T^{5} - 110906046363338 T^{6} + 83374434539 p^{3} T^{7} + 3398738767 p^{6} T^{8} - 3756594 p^{9} T^{9} - 70347 p^{12} T^{10} + 95 p^{15} T^{11} + p^{18} T^{12} \)
37 \( 1 + 262 T - 97404 T^{2} - 9678072 T^{3} + 12194182072 T^{4} + 680381910454 T^{5} - 605701122868778 T^{6} + 680381910454 p^{3} T^{7} + 12194182072 p^{6} T^{8} - 9678072 p^{9} T^{9} - 97404 p^{12} T^{10} + 262 p^{15} T^{11} + p^{18} T^{12} \)
41 \( ( 1 - 244 T + 187983 T^{2} - 33933832 T^{3} + 187983 p^{3} T^{4} - 244 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
43 \( ( 1 - 360 T + 166158 T^{2} - 38975294 T^{3} + 166158 p^{3} T^{4} - 360 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
47 \( 1 - 210 T - 20853 T^{2} + 83809446 T^{3} - 12756928590 T^{4} - 2596137940074 T^{5} + 3698984470026571 T^{6} - 2596137940074 p^{3} T^{7} - 12756928590 p^{6} T^{8} + 83809446 p^{9} T^{9} - 20853 p^{12} T^{10} - 210 p^{15} T^{11} + p^{18} T^{12} \)
53 \( 1 - 393 T - 211446 T^{2} + 23899125 T^{3} + 46453564620 T^{4} + 3425920762143 T^{5} - 9724787230272680 T^{6} + 3425920762143 p^{3} T^{7} + 46453564620 p^{6} T^{8} + 23899125 p^{9} T^{9} - 211446 p^{12} T^{10} - 393 p^{15} T^{11} + p^{18} T^{12} \)
59 \( 1 + 1143 T + 557208 T^{2} + 118327563 T^{3} - 14314666608 T^{4} - 458696646099 p T^{5} - 16891447327378130 T^{6} - 458696646099 p^{4} T^{7} - 14314666608 p^{6} T^{8} + 118327563 p^{9} T^{9} + 557208 p^{12} T^{10} + 1143 p^{15} T^{11} + p^{18} T^{12} \)
61 \( 1 + 70 T - 335143 T^{2} + 129510330 T^{3} + 42145697866 T^{4} - 25171752927730 T^{5} - 316289217432887 T^{6} - 25171752927730 p^{3} T^{7} + 42145697866 p^{6} T^{8} + 129510330 p^{9} T^{9} - 335143 p^{12} T^{10} + 70 p^{15} T^{11} + p^{18} T^{12} \)
67 \( 1 - 628 T - 202942 T^{2} + 436381932 T^{3} - 77667044702 T^{4} - 1097444953952 p T^{5} + 16943668917790 p^{2} T^{6} - 1097444953952 p^{4} T^{7} - 77667044702 p^{6} T^{8} + 436381932 p^{9} T^{9} - 202942 p^{12} T^{10} - 628 p^{15} T^{11} + p^{18} T^{12} \)
71 \( ( 1 + 318 T + 742929 T^{2} + 256167372 T^{3} + 742929 p^{3} T^{4} + 318 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
73 \( 1 - 988 T - 186552 T^{2} + 102237300 T^{3} + 281568890272 T^{4} + 16988127696596 T^{5} - 164639785652996186 T^{6} + 16988127696596 p^{3} T^{7} + 281568890272 p^{6} T^{8} + 102237300 p^{9} T^{9} - 186552 p^{12} T^{10} - 988 p^{15} T^{11} + p^{18} T^{12} \)
79 \( 1 + 861 T - 479895 T^{2} - 258646666 T^{3} + 325257480351 T^{4} - 27564282842211 T^{5} - 246706047980056146 T^{6} - 27564282842211 p^{3} T^{7} + 325257480351 p^{6} T^{8} - 258646666 p^{9} T^{9} - 479895 p^{12} T^{10} + 861 p^{15} T^{11} + p^{18} T^{12} \)
83 \( ( 1 - 519 T + 1583745 T^{2} - 545598870 T^{3} + 1583745 p^{3} T^{4} - 519 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
89 \( 1 + 1766 T + 725929 T^{2} - 728159446 T^{3} - 335534377858 T^{4} + 846551335831238 T^{5} + 1249625385561159997 T^{6} + 846551335831238 p^{3} T^{7} - 335534377858 p^{6} T^{8} - 728159446 p^{9} T^{9} + 725929 p^{12} T^{10} + 1766 p^{15} T^{11} + p^{18} T^{12} \)
97 \( ( 1 + 19 T + 2168419 T^{2} - 10094878 T^{3} + 2168419 p^{3} T^{4} + 19 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
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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

−5.56360517462509607663636028792, −5.52509322311381321164372341732, −5.03610045589832740904822569056, −4.85891579313912186079242334992, −4.76927284604221326219772686261, −4.69115469685437902619954024268, −4.33403753550246576633681050488, −4.33369932422485648256194479638, −4.08358416423898795554591788345, −4.04584860930676832018346688696, −3.68318422021775237385144339894, −3.47108189991406044327049481374, −3.32710009159538161233182162359, −2.81756153933936493034272681450, −2.73216010362568910415006088164, −2.71524187246411102586192554592, −2.51333192088507661411966338257, −2.30706017889915223595533478757, −1.89628411731295137654806048494, −1.65221613018659013127719043014, −1.43364985558437990658084809992, −0.839381615845157463671449478460, −0.819136056180161874467411004213, −0.35321346059477544382161028521, −0.01729676698181807466941693272, 0.01729676698181807466941693272, 0.35321346059477544382161028521, 0.819136056180161874467411004213, 0.839381615845157463671449478460, 1.43364985558437990658084809992, 1.65221613018659013127719043014, 1.89628411731295137654806048494, 2.30706017889915223595533478757, 2.51333192088507661411966338257, 2.71524187246411102586192554592, 2.73216010362568910415006088164, 2.81756153933936493034272681450, 3.32710009159538161233182162359, 3.47108189991406044327049481374, 3.68318422021775237385144339894, 4.04584860930676832018346688696, 4.08358416423898795554591788345, 4.33369932422485648256194479638, 4.33403753550246576633681050488, 4.69115469685437902619954024268, 4.76927284604221326219772686261, 4.85891579313912186079242334992, 5.03610045589832740904822569056, 5.52509322311381321164372341732, 5.56360517462509607663636028792

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

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