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Properties

Label 12-2673e6-1.1-c1e6-0-0
Degree $12$
Conductor $3.647\times 10^{20}$
Sign $1$
Analytic cond. $9.45488\times 10^{7}$
Root an. cond. $4.61995$
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  + 2·2-s + 2·5-s − 2·7-s − 2·8-s + 4·10-s + 6·11-s − 8·13-s − 4·14-s − 2·16-s + 2·17-s − 4·19-s + 12·22-s + 10·23-s − 9·25-s − 16·26-s + 6·29-s − 8·31-s − 4·32-s + 4·34-s − 4·35-s + 2·37-s − 8·38-s − 4·40-s − 4·41-s + 2·43-s + 20·46-s + 28·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.894·5-s − 0.755·7-s − 0.707·8-s + 1.26·10-s + 1.80·11-s − 2.21·13-s − 1.06·14-s − 1/2·16-s + 0.485·17-s − 0.917·19-s + 2.55·22-s + 2.08·23-s − 9/5·25-s − 3.13·26-s + 1.11·29-s − 1.43·31-s − 0.707·32-s + 0.685·34-s − 0.676·35-s + 0.328·37-s − 1.29·38-s − 0.632·40-s − 0.624·41-s + 0.304·43-s + 2.94·46-s + 4.08·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{30} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(9.45488\times 10^{7}\)
Root analytic conductor: \(4.61995\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{30} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.950795574\)
\(L(\frac12)\) \(\approx\) \(7.950795574\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( ( 1 - T )^{6} \)
good2 \( 1 - p T + p^{2} T^{2} - 3 p T^{3} + 5 p T^{4} - 3 p^{2} T^{5} + 15 T^{6} - 3 p^{3} T^{7} + 5 p^{3} T^{8} - 3 p^{4} T^{9} + p^{6} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 2 T + 13 T^{2} - 36 T^{3} + 112 T^{4} - 282 T^{5} + 672 T^{6} - 282 p T^{7} + 112 p^{2} T^{8} - 36 p^{3} T^{9} + 13 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 2 T + 20 T^{2} + 10 T^{3} + 195 T^{4} + 60 T^{5} + 1707 T^{6} + 60 p T^{7} + 195 p^{2} T^{8} + 10 p^{3} T^{9} + 20 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 8 T + 50 T^{2} + 16 p T^{3} + 633 T^{4} + 1398 T^{5} + 3951 T^{6} + 1398 p T^{7} + 633 p^{2} T^{8} + 16 p^{4} T^{9} + 50 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 2 T + 49 T^{2} - 138 T^{3} + 1354 T^{4} - 3708 T^{5} + 27408 T^{6} - 3708 p T^{7} + 1354 p^{2} T^{8} - 138 p^{3} T^{9} + 49 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 4 T + 63 T^{2} + 128 T^{3} + 90 p T^{4} + 1804 T^{5} + 34859 T^{6} + 1804 p T^{7} + 90 p^{3} T^{8} + 128 p^{3} T^{9} + 63 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 10 T + 85 T^{2} - 576 T^{3} + 3256 T^{4} - 16398 T^{5} + 86436 T^{6} - 16398 p T^{7} + 3256 p^{2} T^{8} - 576 p^{3} T^{9} + 85 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 6 T + 129 T^{2} - 678 T^{3} + 7470 T^{4} - 33996 T^{5} + 265024 T^{6} - 33996 p T^{7} + 7470 p^{2} T^{8} - 678 p^{3} T^{9} + 129 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 8 T + 74 T^{2} + 658 T^{3} + 3687 T^{4} + 20706 T^{5} + 146229 T^{6} + 20706 p T^{7} + 3687 p^{2} T^{8} + 658 p^{3} T^{9} + 74 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 2 T + 69 T^{2} - 46 T^{3} + 90 p T^{4} + 1510 T^{5} + 109901 T^{6} + 1510 p T^{7} + 90 p^{3} T^{8} - 46 p^{3} T^{9} + 69 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 4 T + 118 T^{2} + 276 T^{3} + 6463 T^{4} + 5448 T^{5} + 261876 T^{6} + 5448 p T^{7} + 6463 p^{2} T^{8} + 276 p^{3} T^{9} + 118 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 2 T + 111 T^{2} + 50 T^{3} + 7227 T^{4} + 12172 T^{5} + 329690 T^{6} + 12172 p T^{7} + 7227 p^{2} T^{8} + 50 p^{3} T^{9} + 111 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 28 T + 544 T^{2} - 7320 T^{3} + 80392 T^{4} - 709908 T^{5} + 5346798 T^{6} - 709908 p T^{7} + 80392 p^{2} T^{8} - 7320 p^{3} T^{9} + 544 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 24 T + 456 T^{2} - 6072 T^{3} + 68592 T^{4} - 629688 T^{5} + 5009470 T^{6} - 629688 p T^{7} + 68592 p^{2} T^{8} - 6072 p^{3} T^{9} + 456 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 8 T + 142 T^{2} + 1272 T^{3} + 14071 T^{4} + 112656 T^{5} + 937956 T^{6} + 112656 p T^{7} + 14071 p^{2} T^{8} + 1272 p^{3} T^{9} + 142 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 2 T + 246 T^{2} - 928 T^{3} + 28458 T^{4} - 123962 T^{5} + 2103182 T^{6} - 123962 p T^{7} + 28458 p^{2} T^{8} - 928 p^{3} T^{9} + 246 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 12 T + 396 T^{2} - 3568 T^{3} + 64692 T^{4} - 450012 T^{5} + 5721318 T^{6} - 450012 p T^{7} + 64692 p^{2} T^{8} - 3568 p^{3} T^{9} + 396 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 30 T + 621 T^{2} - 9276 T^{3} + 117528 T^{4} - 1231254 T^{5} + 11277688 T^{6} - 1231254 p T^{7} + 117528 p^{2} T^{8} - 9276 p^{3} T^{9} + 621 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 16 T + 278 T^{2} - 3008 T^{3} + 33087 T^{4} - 298224 T^{5} + 2686836 T^{6} - 298224 p T^{7} + 33087 p^{2} T^{8} - 3008 p^{3} T^{9} + 278 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 12 T + 447 T^{2} + 4280 T^{3} + 85686 T^{4} + 643428 T^{5} + 8939739 T^{6} + 643428 p T^{7} + 85686 p^{2} T^{8} + 4280 p^{3} T^{9} + 447 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 207 T^{2} + 558 T^{3} + 25038 T^{4} + 117054 T^{5} + 2197384 T^{6} + 117054 p T^{7} + 25038 p^{2} T^{8} + 558 p^{3} T^{9} + 207 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 + 6 T + 477 T^{2} + 2544 T^{3} + 99408 T^{4} + 441714 T^{5} + 11545804 T^{6} + 441714 p T^{7} + 99408 p^{2} T^{8} + 2544 p^{3} T^{9} + 477 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 18 T + 483 T^{2} + 4286 T^{3} + 64239 T^{4} + 263040 T^{5} + 5020554 T^{6} + 263040 p T^{7} + 64239 p^{2} T^{8} + 4286 p^{3} T^{9} + 483 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.67553530568212578512418077476, −4.35737061276011472392725425729, −4.28177718457579247728788998970, −4.09109534582717484996888725842, −3.93391668220929779614302378018, −3.93060695663565604008898817820, −3.87174012180955390326465297366, −3.53995297976880907050642849564, −3.45902138812817361265239287915, −3.45220897119758923309240678282, −3.08952464304058762192861292005, −2.91314097566522839587989260777, −2.63752664541325893307370286157, −2.53423174012063045820957303923, −2.53312788957420812509891886907, −2.31908177535734800993094185149, −2.03496664942275407703111535721, −1.92952035541475031907062972008, −1.76239634791823541166389407612, −1.60788511197874836144326695752, −1.25104564343905959095929477919, −0.76041174591612154266939318243, −0.75108250598246465199308883282, −0.74833888982828642016509566297, −0.22079226523069507188313970314, 0.22079226523069507188313970314, 0.74833888982828642016509566297, 0.75108250598246465199308883282, 0.76041174591612154266939318243, 1.25104564343905959095929477919, 1.60788511197874836144326695752, 1.76239634791823541166389407612, 1.92952035541475031907062972008, 2.03496664942275407703111535721, 2.31908177535734800993094185149, 2.53312788957420812509891886907, 2.53423174012063045820957303923, 2.63752664541325893307370286157, 2.91314097566522839587989260777, 3.08952464304058762192861292005, 3.45220897119758923309240678282, 3.45902138812817361265239287915, 3.53995297976880907050642849564, 3.87174012180955390326465297366, 3.93060695663565604008898817820, 3.93391668220929779614302378018, 4.09109534582717484996888725842, 4.28177718457579247728788998970, 4.35737061276011472392725425729, 4.67553530568212578512418077476

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

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