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Properties

Label 12-722e6-1.1-c1e6-0-10
Degree $12$
Conductor $1.417\times 10^{17}$
Sign $1$
Analytic cond. $36718.5$
Root an. cond. $2.40108$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 6·7-s + 8-s + 3·9-s − 6·11-s + 6·13-s + 24·17-s − 18·21-s + 6·23-s + 3·24-s − 27-s − 18·29-s − 6·31-s − 18·33-s + 12·37-s + 18·39-s + 6·41-s + 12·43-s − 24·47-s + 21·49-s + 72·51-s − 6·53-s − 6·56-s + 12·59-s − 12·61-s − 18·63-s + 9·67-s + ⋯
L(s)  = 1  + 1.73·3-s − 2.26·7-s + 0.353·8-s + 9-s − 1.80·11-s + 1.66·13-s + 5.82·17-s − 3.92·21-s + 1.25·23-s + 0.612·24-s − 0.192·27-s − 3.34·29-s − 1.07·31-s − 3.13·33-s + 1.97·37-s + 2.88·39-s + 0.937·41-s + 1.82·43-s − 3.50·47-s + 3·49-s + 10.0·51-s − 0.824·53-s − 0.801·56-s + 1.56·59-s − 1.53·61-s − 2.26·63-s + 1.09·67-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(36718.5\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.016911831\)
\(L(\frac12)\) \(\approx\) \(7.016911831\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
19 \( 1 \)
good3 \( 1 - p T + 2 p T^{2} - 8 T^{3} + p T^{4} + 7 p T^{5} - 53 T^{6} + 7 p^{2} T^{7} + p^{3} T^{8} - 8 p^{3} T^{9} + 2 p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 22 T^{3} + 359 T^{6} - 22 p^{3} T^{9} + p^{6} T^{12} \)
7 \( 1 + 6 T + 15 T^{2} + 6 T^{3} - 66 T^{4} - 30 p T^{5} - 565 T^{6} - 30 p^{2} T^{7} - 66 p^{2} T^{8} + 6 p^{3} T^{9} + 15 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 6 T - 10 T^{3} + 222 T^{4} + 42 T^{5} - 3181 T^{6} + 42 p T^{7} + 222 p^{2} T^{8} - 10 p^{3} T^{9} + 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 6 T + 48 T^{2} - 230 T^{3} + 1332 T^{4} - 4950 T^{5} + 21111 T^{6} - 4950 p T^{7} + 1332 p^{2} T^{8} - 230 p^{3} T^{9} + 48 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 24 T + 252 T^{2} - 1458 T^{3} + 4356 T^{4} - 852 T^{5} - 34973 T^{6} - 852 p T^{7} + 4356 p^{2} T^{8} - 1458 p^{3} T^{9} + 252 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T + 24 T^{2} - 64 T^{3} + 600 T^{4} - 4734 T^{5} + 22673 T^{6} - 4734 p T^{7} + 600 p^{2} T^{8} - 64 p^{3} T^{9} + 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 18 T + 216 T^{2} + 2012 T^{3} + 540 p T^{4} + 103320 T^{5} + 592055 T^{6} + 103320 p T^{7} + 540 p^{3} T^{8} + 2012 p^{3} T^{9} + 216 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 6 T - 33 T^{2} - 346 T^{3} + 342 T^{4} + 6318 T^{5} + 21795 T^{6} + 6318 p T^{7} + 342 p^{2} T^{8} - 346 p^{3} T^{9} - 33 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 6 T + 87 T^{2} - 308 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 6 T + 15 T^{2} - 19 T^{3} - 111 T^{4} + 5409 T^{5} - 84418 T^{6} + 5409 p T^{7} - 111 p^{2} T^{8} - 19 p^{3} T^{9} + 15 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 12 T + 78 T^{2} - 656 T^{3} + 3366 T^{4} - 9846 T^{5} + 57393 T^{6} - 9846 p T^{7} + 3366 p^{2} T^{8} - 656 p^{3} T^{9} + 78 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 24 T + 336 T^{2} + 3530 T^{3} + 33492 T^{4} + 280062 T^{5} + 2072849 T^{6} + 280062 p T^{7} + 33492 p^{2} T^{8} + 3530 p^{3} T^{9} + 336 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 6 T + 96 T^{2} + 640 T^{3} + 5064 T^{4} + 38970 T^{5} + 247403 T^{6} + 38970 p T^{7} + 5064 p^{2} T^{8} + 640 p^{3} T^{9} + 96 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 12 T + 63 T^{2} - 405 T^{3} - 1557 T^{4} + 56967 T^{5} - 487862 T^{6} + 56967 p T^{7} - 1557 p^{2} T^{8} - 405 p^{3} T^{9} + 63 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 12 T + 24 T^{2} - 1018 T^{3} - 7596 T^{4} + 16866 T^{5} + 630603 T^{6} + 16866 p T^{7} - 7596 p^{2} T^{8} - 1018 p^{3} T^{9} + 24 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 9 T - 396 T^{3} - 2511 T^{4} + 477 p T^{5} + 308195 T^{6} + 477 p^{2} T^{7} - 2511 p^{2} T^{8} - 396 p^{3} T^{9} - 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 36 T^{2} - 572 T^{3} - 4788 T^{4} + 50202 T^{5} + 489077 T^{6} + 50202 p T^{7} - 4788 p^{2} T^{8} - 572 p^{3} T^{9} - 36 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 + 3 T + 234 T^{2} + 1092 T^{3} + 30069 T^{4} + 154137 T^{5} + 2558249 T^{6} + 154137 p T^{7} + 30069 p^{2} T^{8} + 1092 p^{3} T^{9} + 234 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 6 T + 60 T^{2} - 782 T^{3} + 2952 T^{4} - 9108 T^{5} + 924705 T^{6} - 9108 p T^{7} + 2952 p^{2} T^{8} - 782 p^{3} T^{9} + 60 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 6 T - 186 T^{2} - 558 T^{3} + 25188 T^{4} + 30012 T^{5} - 2301977 T^{6} + 30012 p T^{7} + 25188 p^{2} T^{8} - 558 p^{3} T^{9} - 186 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 36 T + 576 T^{2} - 4698 T^{3} + 4932 T^{4} + 412812 T^{5} - 5872661 T^{6} + 412812 p T^{7} + 4932 p^{2} T^{8} - 4698 p^{3} T^{9} + 576 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 3 T + 6 T^{2} + 8 T^{3} - 279 T^{4} - 55575 T^{5} - 995607 T^{6} - 55575 p T^{7} - 279 p^{2} T^{8} + 8 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.68095224289561880762332290194, −5.39534484462305093268567895381, −5.18928517497906694022749535111, −5.18125912546448353278085996483, −4.90959213435018198480283951784, −4.85360237016071960798393965691, −4.53324026091079419348601989242, −4.15109462074128212412604614254, −3.87792855627997917769307554239, −3.72427634512559210856515651019, −3.60043081364705621982157028321, −3.54950764936458945228400640939, −3.53824737540651579463443417124, −3.24151928754895181241788323200, −3.15352244770969710165798345542, −2.76978435945731810225636066002, −2.70849723267534582937803739615, −2.58208713847238643188167485476, −2.47271376232298767509749384178, −1.76922251026558016932038086795, −1.69735858448953273931053739862, −1.50318125512993839454408267957, −1.12304152330318440944137336798, −0.70703105802893632091941364050, −0.50409687910883025561310166775, 0.50409687910883025561310166775, 0.70703105802893632091941364050, 1.12304152330318440944137336798, 1.50318125512993839454408267957, 1.69735858448953273931053739862, 1.76922251026558016932038086795, 2.47271376232298767509749384178, 2.58208713847238643188167485476, 2.70849723267534582937803739615, 2.76978435945731810225636066002, 3.15352244770969710165798345542, 3.24151928754895181241788323200, 3.53824737540651579463443417124, 3.54950764936458945228400640939, 3.60043081364705621982157028321, 3.72427634512559210856515651019, 3.87792855627997917769307554239, 4.15109462074128212412604614254, 4.53324026091079419348601989242, 4.85360237016071960798393965691, 4.90959213435018198480283951784, 5.18125912546448353278085996483, 5.18928517497906694022749535111, 5.39534484462305093268567895381, 5.68095224289561880762332290194

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

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