Properties

Label 12-5054e6-1.1-c1e6-0-5
Degree $12$
Conductor $1.667\times 10^{22}$
Sign $1$
Analytic cond. $4.31990\times 10^{9}$
Root an. cond. $6.35266$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s − 3·3-s + 21·4-s − 3·5-s − 18·6-s − 6·7-s + 56·8-s − 6·9-s − 18·10-s + 3·11-s − 63·12-s − 6·13-s − 36·14-s + 9·15-s + 126·16-s − 9·17-s − 36·18-s − 63·20-s + 18·21-s + 18·22-s − 168·24-s − 9·25-s − 36·26-s + 29·27-s − 126·28-s − 6·29-s + 54·30-s + ⋯
L(s)  = 1  + 4.24·2-s − 1.73·3-s + 21/2·4-s − 1.34·5-s − 7.34·6-s − 2.26·7-s + 19.7·8-s − 2·9-s − 5.69·10-s + 0.904·11-s − 18.1·12-s − 1.66·13-s − 9.62·14-s + 2.32·15-s + 63/2·16-s − 2.18·17-s − 8.48·18-s − 14.0·20-s + 3.92·21-s + 3.83·22-s − 34.2·24-s − 9/5·25-s − 7.06·26-s + 5.58·27-s − 23.8·28-s − 1.11·29-s + 9.85·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 7^{6} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(4.31990\times 10^{9}\)
Root analytic conductor: \(6.35266\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 7^{6} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{6} \)
7 \( ( 1 + T )^{6} \)
19 \( 1 \)
good3 \( 1 + p T + 5 p T^{2} + 34 T^{3} + 34 p T^{4} + 20 p^{2} T^{5} + 395 T^{6} + 20 p^{3} T^{7} + 34 p^{3} T^{8} + 34 p^{3} T^{9} + 5 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 + 3 T + 18 T^{2} + 28 T^{3} + 93 T^{4} + 9 p T^{5} + 288 T^{6} + 9 p^{2} T^{7} + 93 p^{2} T^{8} + 28 p^{3} T^{9} + 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T + 57 T^{2} - 150 T^{3} + 12 p^{2} T^{4} - 3120 T^{5} + 20813 T^{6} - 3120 p T^{7} + 12 p^{4} T^{8} - 150 p^{3} T^{9} + 57 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 6 T + 81 T^{2} + 373 T^{3} + 2679 T^{4} + 9489 T^{5} + 46678 T^{6} + 9489 p T^{7} + 2679 p^{2} T^{8} + 373 p^{3} T^{9} + 81 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 9 T + 78 T^{2} + 416 T^{3} + 1965 T^{4} + 7863 T^{5} + 31553 T^{6} + 7863 p T^{7} + 1965 p^{2} T^{8} + 416 p^{3} T^{9} + 78 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 78 T^{2} + 144 T^{3} + 2895 T^{4} + 8208 T^{5} + 76196 T^{6} + 8208 p T^{7} + 2895 p^{2} T^{8} + 144 p^{3} T^{9} + 78 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 + 6 T + 147 T^{2} + 605 T^{3} + 8925 T^{4} + 27249 T^{5} + 319470 T^{6} + 27249 p T^{7} + 8925 p^{2} T^{8} + 605 p^{3} T^{9} + 147 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T + 114 T^{2} + 242 T^{3} + 6651 T^{4} + 12411 T^{5} + 255204 T^{6} + 12411 p T^{7} + 6651 p^{2} T^{8} + 242 p^{3} T^{9} + 114 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T + 120 T^{2} - 522 T^{3} + 7149 T^{4} - 38031 T^{5} + 8148 p T^{6} - 38031 p T^{7} + 7149 p^{2} T^{8} - 522 p^{3} T^{9} + 120 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 9 T + 237 T^{2} + 1692 T^{3} + 23622 T^{4} + 132768 T^{5} + 1278061 T^{6} + 132768 p T^{7} + 23622 p^{2} T^{8} + 1692 p^{3} T^{9} + 237 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 6 T + 3 p T^{2} + 558 T^{3} + 9786 T^{4} + 33942 T^{5} + 479649 T^{6} + 33942 p T^{7} + 9786 p^{2} T^{8} + 558 p^{3} T^{9} + 3 p^{5} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 9 T + 162 T^{2} - 756 T^{3} + 6651 T^{4} - 4491 T^{5} + 118988 T^{6} - 4491 p T^{7} + 6651 p^{2} T^{8} - 756 p^{3} T^{9} + 162 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 6 T + 171 T^{2} + 463 T^{3} + 13755 T^{4} + 23175 T^{5} + 844938 T^{6} + 23175 p T^{7} + 13755 p^{2} T^{8} + 463 p^{3} T^{9} + 171 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 15 T + 309 T^{2} + 3010 T^{3} + 630 p T^{4} + 281676 T^{5} + 2687447 T^{6} + 281676 p T^{7} + 630 p^{3} T^{8} + 3010 p^{3} T^{9} + 309 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 30 T + 570 T^{2} + 7798 T^{3} + 88647 T^{4} + 842676 T^{5} + 7062604 T^{6} + 842676 p T^{7} + 88647 p^{2} T^{8} + 7798 p^{3} T^{9} + 570 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 15 T + 309 T^{2} + 3452 T^{3} + 45504 T^{4} + 397800 T^{5} + 3877017 T^{6} + 397800 p T^{7} + 45504 p^{2} T^{8} + 3452 p^{3} T^{9} + 309 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 21 T + 402 T^{2} + 4866 T^{3} + 58551 T^{4} + 529845 T^{5} + 4986884 T^{6} + 529845 p T^{7} + 58551 p^{2} T^{8} + 4866 p^{3} T^{9} + 402 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 33 T + 9 p T^{2} + 9652 T^{3} + 116484 T^{4} + 1197894 T^{5} + 148711 p T^{6} + 1197894 p T^{7} + 116484 p^{2} T^{8} + 9652 p^{3} T^{9} + 9 p^{5} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 30 T + 657 T^{2} - 135 p T^{3} + 143751 T^{4} - 1602123 T^{5} + 15515766 T^{6} - 1602123 p T^{7} + 143751 p^{2} T^{8} - 135 p^{4} T^{9} + 657 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 33 T + 711 T^{2} + 11582 T^{3} + 154680 T^{4} + 1739418 T^{5} + 17033639 T^{6} + 1739418 p T^{7} + 154680 p^{2} T^{8} + 11582 p^{3} T^{9} + 711 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 12 T + 399 T^{2} - 4644 T^{3} + 76542 T^{4} - 761340 T^{5} + 8708443 T^{6} - 761340 p T^{7} + 76542 p^{2} T^{8} - 4644 p^{3} T^{9} + 399 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 18 T + 405 T^{2} - 6733 T^{3} + 90915 T^{4} - 1078941 T^{5} + 11856478 T^{6} - 1078941 p T^{7} + 90915 p^{2} T^{8} - 6733 p^{3} T^{9} + 405 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.73990619842807196505790531382, −4.54531571956762994167228742227, −4.30313311676829108861910127123, −4.16614504037790056231574439662, −4.05990881811628495078579179554, −3.98755545247277190612902846882, −3.90390710291417826457541748366, −3.62835674561288254009491494097, −3.59811008165312505320782217550, −3.54379697724643340091556050364, −3.31239088862591286654256706906, −3.18056380266124993101865190320, −3.06493299994889812145115947459, −2.73180515775055965351274684465, −2.64907564146307708998934846692, −2.64175511537901909327376510859, −2.54007309680448523933632496116, −2.51544887755557943042221831110, −2.46722402798529770409854248614, −1.79922562032062022574446335893, −1.73784506221736067375986294003, −1.67028858322732445686296007562, −1.49271060918704932929773731450, −1.16192254524391169954681945871, −1.06369462285848912348211704325, 0, 0, 0, 0, 0, 0, 1.06369462285848912348211704325, 1.16192254524391169954681945871, 1.49271060918704932929773731450, 1.67028858322732445686296007562, 1.73784506221736067375986294003, 1.79922562032062022574446335893, 2.46722402798529770409854248614, 2.51544887755557943042221831110, 2.54007309680448523933632496116, 2.64175511537901909327376510859, 2.64907564146307708998934846692, 2.73180515775055965351274684465, 3.06493299994889812145115947459, 3.18056380266124993101865190320, 3.31239088862591286654256706906, 3.54379697724643340091556050364, 3.59811008165312505320782217550, 3.62835674561288254009491494097, 3.90390710291417826457541748366, 3.98755545247277190612902846882, 4.05990881811628495078579179554, 4.16614504037790056231574439662, 4.30313311676829108861910127123, 4.54531571956762994167228742227, 4.73990619842807196505790531382

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

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