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Properties

Label 12-2888e6-1.1-c1e6-0-0
Degree $12$
Conductor $5.802\times 10^{20}$
Sign $1$
Analytic cond. $1.50399\times 10^{8}$
Root an. cond. $4.80216$
Motivic weight $1$
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  + 3·3-s + 2·5-s − 2·7-s − 5·11-s + 10·13-s + 6·15-s + 4·17-s − 6·21-s − 15·23-s − 7·25-s − 6·27-s − 7·29-s + 5·31-s − 15·33-s − 4·35-s + 17·37-s + 30·39-s + 4·41-s − 11·43-s + 18·47-s − 9·49-s + 12·51-s − 7·53-s − 10·55-s + 59-s − 4·61-s + 20·65-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.894·5-s − 0.755·7-s − 1.50·11-s + 2.77·13-s + 1.54·15-s + 0.970·17-s − 1.30·21-s − 3.12·23-s − 7/5·25-s − 1.15·27-s − 1.29·29-s + 0.898·31-s − 2.61·33-s − 0.676·35-s + 2.79·37-s + 4.80·39-s + 0.624·41-s − 1.67·43-s + 2.62·47-s − 9/7·49-s + 1.68·51-s − 0.961·53-s − 1.34·55-s + 0.130·59-s − 0.512·61-s + 2.48·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(1.50399\times 10^{8}\)
Root analytic conductor: \(4.80216\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.73123089\)
\(L(\frac12)\) \(\approx\) \(10.73123089\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 - p T + p^{2} T^{2} - 7 p T^{3} + 17 p T^{4} - 94 T^{5} + 182 T^{6} - 94 p T^{7} + 17 p^{3} T^{8} - 7 p^{4} T^{9} + p^{6} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 2 T + 11 T^{2} - 24 T^{3} + 104 T^{4} - 36 p T^{5} + 544 T^{6} - 36 p^{2} T^{7} + 104 p^{2} T^{8} - 24 p^{3} T^{9} + 11 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 2 T + 13 T^{2} + 32 T^{3} + 135 T^{4} + 222 T^{5} + 998 T^{6} + 222 p T^{7} + 135 p^{2} T^{8} + 32 p^{3} T^{9} + 13 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 5 T + 37 T^{2} + 109 T^{3} + 423 T^{4} + 724 T^{5} + 2998 T^{6} + 724 p T^{7} + 423 p^{2} T^{8} + 109 p^{3} T^{9} + 37 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 10 T + 84 T^{2} - 456 T^{3} + 2161 T^{4} - 8512 T^{5} + 31877 T^{6} - 8512 p T^{7} + 2161 p^{2} T^{8} - 456 p^{3} T^{9} + 84 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 4 T + 33 T^{2} - 200 T^{3} + 956 T^{4} - 4286 T^{5} + 23180 T^{6} - 4286 p T^{7} + 956 p^{2} T^{8} - 200 p^{3} T^{9} + 33 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 15 T + 165 T^{2} + 1133 T^{3} + 6635 T^{4} + 30190 T^{5} + 148750 T^{6} + 30190 p T^{7} + 6635 p^{2} T^{8} + 1133 p^{3} T^{9} + 165 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 7 T + 52 T^{2} - 30 T^{3} + 497 T^{4} + 22 p T^{5} + 53681 T^{6} + 22 p^{2} T^{7} + 497 p^{2} T^{8} - 30 p^{3} T^{9} + 52 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 5 T + 131 T^{2} - 475 T^{3} + 7995 T^{4} - 23150 T^{5} + 305490 T^{6} - 23150 p T^{7} + 7995 p^{2} T^{8} - 475 p^{3} T^{9} + 131 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 17 T + 262 T^{2} - 2770 T^{3} + 25315 T^{4} - 191782 T^{5} + 1257639 T^{6} - 191782 p T^{7} + 25315 p^{2} T^{8} - 2770 p^{3} T^{9} + 262 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 4 T + 130 T^{2} - 666 T^{3} + 8983 T^{4} - 51438 T^{5} + 425177 T^{6} - 51438 p T^{7} + 8983 p^{2} T^{8} - 666 p^{3} T^{9} + 130 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 11 T + 105 T^{2} + 1045 T^{3} + 8203 T^{4} + 58174 T^{5} + 411206 T^{6} + 58174 p T^{7} + 8203 p^{2} T^{8} + 1045 p^{3} T^{9} + 105 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 18 T + 7 p T^{2} - 3816 T^{3} + 40883 T^{4} - 339102 T^{5} + 2597622 T^{6} - 339102 p T^{7} + 40883 p^{2} T^{8} - 3816 p^{3} T^{9} + 7 p^{5} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 7 T + 216 T^{2} + 1670 T^{3} + 23205 T^{4} + 163178 T^{5} + 1545617 T^{6} + 163178 p T^{7} + 23205 p^{2} T^{8} + 1670 p^{3} T^{9} + 216 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - T + 185 T^{2} + 7 T^{3} + 16803 T^{4} + 20452 T^{5} + 1104278 T^{6} + 20452 p T^{7} + 16803 p^{2} T^{8} + 7 p^{3} T^{9} + 185 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 4 T + 274 T^{2} + 1052 T^{3} + 35199 T^{4} + 115084 T^{5} + 2710451 T^{6} + 115084 p T^{7} + 35199 p^{2} T^{8} + 1052 p^{3} T^{9} + 274 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 20 T + 445 T^{2} - 5656 T^{3} + 73075 T^{4} - 683740 T^{5} + 6411630 T^{6} - 683740 p T^{7} + 73075 p^{2} T^{8} - 5656 p^{3} T^{9} + 445 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 6 T + 293 T^{2} + 1300 T^{3} + 39227 T^{4} + 139654 T^{5} + 3349774 T^{6} + 139654 p T^{7} + 39227 p^{2} T^{8} + 1300 p^{3} T^{9} + 293 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 2 T + 400 T^{2} + 778 T^{3} + 68925 T^{4} + 116918 T^{5} + 6577955 T^{6} + 116918 p T^{7} + 68925 p^{2} T^{8} + 778 p^{3} T^{9} + 400 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 8 T + 358 T^{2} - 2560 T^{3} + 61487 T^{4} - 367992 T^{5} + 6170964 T^{6} - 367992 p T^{7} + 61487 p^{2} T^{8} - 2560 p^{3} T^{9} + 358 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 22 T + 510 T^{2} + 6834 T^{3} + 90471 T^{4} + 899140 T^{5} + 9093956 T^{6} + 899140 p T^{7} + 90471 p^{2} T^{8} + 6834 p^{3} T^{9} + 510 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 14 T + 496 T^{2} - 5974 T^{3} + 105729 T^{4} - 1041670 T^{5} + 12370299 T^{6} - 1041670 p T^{7} + 105729 p^{2} T^{8} - 5974 p^{3} T^{9} + 496 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 41 T + 926 T^{2} - 14554 T^{3} + 184359 T^{4} - 2018114 T^{5} + 20490063 T^{6} - 2018114 p T^{7} + 184359 p^{2} T^{8} - 14554 p^{3} T^{9} + 926 p^{4} T^{10} - 41 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.42879823299282210686731380284, −4.21804146679736704018392303658, −4.20399499439474149170107384347, −4.13807930348442638189835170177, −3.91622901959441559208131231204, −3.69429101551770559559255788241, −3.58714888727795413600857877186, −3.32695835051530302361082071948, −3.21266442952261330584478907275, −3.20656144986804790277435272873, −3.07951663116581207638682653899, −3.00960168805064195539026222215, −2.88865402562044063757349037555, −2.35540086961771203370108501042, −2.17218109573658863277235516895, −2.09748587867115481274653935223, −2.09287322719689974222647382170, −2.07436445907718846361317595697, −2.01819972206051935000739945871, −1.44990213045023542957495409096, −1.19039526031485848625398454332, −1.06132744313796677867147785709, −0.78322896313463279545759471942, −0.44195226567705230577551452762, −0.32238529753729888030919416778, 0.32238529753729888030919416778, 0.44195226567705230577551452762, 0.78322896313463279545759471942, 1.06132744313796677867147785709, 1.19039526031485848625398454332, 1.44990213045023542957495409096, 2.01819972206051935000739945871, 2.07436445907718846361317595697, 2.09287322719689974222647382170, 2.09748587867115481274653935223, 2.17218109573658863277235516895, 2.35540086961771203370108501042, 2.88865402562044063757349037555, 3.00960168805064195539026222215, 3.07951663116581207638682653899, 3.20656144986804790277435272873, 3.21266442952261330584478907275, 3.32695835051530302361082071948, 3.58714888727795413600857877186, 3.69429101551770559559255788241, 3.91622901959441559208131231204, 4.13807930348442638189835170177, 4.20399499439474149170107384347, 4.21804146679736704018392303658, 4.42879823299282210686731380284

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

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