Properties

Label 12-63e6-1.1-c3e6-0-0
Degree $12$
Conductor $62523502209$
Sign $1$
Analytic cond. $2637.78$
Root an. cond. $1.92798$
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 − 13·7-s − 21·8-s + 11·10-s + 35·11-s + 124·13-s − 13·14-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 − 143·35-s − 262·37-s + 202·38-s − 231·40-s − 488·41-s + 720·43-s + 216·46-s − 210·47-s + ⋯
L(s)  = 1  + 0.353·2-s + 0.983·5-s − 0.701·7-s − 0.928·8-s + 0.347·10-s + 0.959·11-s + 2.64·13-s − 0.248·14-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 − 0.690·35-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 + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(2637.78\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{63} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(7.454464530\)
\(L(\frac12)\) \(\approx\) \(7.454464530\)
\(L(\frac{5}{2})\) 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 \( 1 + 13 T + 236 T^{2} + 1735 p T^{3} + 236 p^{3} T^{4} + 13 p^{6} T^{5} + p^{9} T^{6} \)
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

−8.436629753000804855150440155583, −7.53462422835805311758776901408, −7.43602404321788974473835527392, −7.35684776850756491610247421500, −7.12034874804432587392200085234, −6.76086857673915214559187784528, −6.60096639176347016691888443397, −6.38646242417868593798458354133, −5.95496696203091485497266758813, −5.83666597737119012847344002842, −5.78240048439006285499258704692, −5.58385534520113376740187272825, −5.05044062002673735222264051969, −4.82826556312930327921916471797, −4.77168194223381093535637212297, −4.06283711919455872711754429935, −3.67537473826489377957443694380, −3.65401194276581920470012029282, −3.13634886043002296919491938856, −3.13314450190814193489436178205, −2.79506605545043200863011374633, −2.07171678858862665835590030403, −1.46787699426438346279439529441, −0.970356283978368842302507435992, −0.890307964620169678509682771276, 0.890307964620169678509682771276, 0.970356283978368842302507435992, 1.46787699426438346279439529441, 2.07171678858862665835590030403, 2.79506605545043200863011374633, 3.13314450190814193489436178205, 3.13634886043002296919491938856, 3.65401194276581920470012029282, 3.67537473826489377957443694380, 4.06283711919455872711754429935, 4.77168194223381093535637212297, 4.82826556312930327921916471797, 5.05044062002673735222264051969, 5.58385534520113376740187272825, 5.78240048439006285499258704692, 5.83666597737119012847344002842, 5.95496696203091485497266758813, 6.38646242417868593798458354133, 6.60096639176347016691888443397, 6.76086857673915214559187784528, 7.12034874804432587392200085234, 7.35684776850756491610247421500, 7.43602404321788974473835527392, 7.53462422835805311758776901408, 8.436629753000804855150440155583

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

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