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Properties

Label 12-567e6-1.1-c1e6-0-3
Degree $12$
Conductor $3.323\times 10^{16}$
Sign $1$
Analytic cond. $8613.12$
Root an. cond. $2.12779$
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  − 2·2-s + 3·4-s + 2·5-s + 2·7-s − 4·10-s + 14·11-s − 2·13-s − 4·14-s − 4·16-s − 5·19-s + 6·20-s − 28·22-s + 12·23-s − 15·25-s + 4·26-s + 6·28-s − 13·29-s + 8·31-s + 7·32-s + 4·35-s + 8·37-s + 10·38-s − 2·41-s + 9·43-s + 42·44-s − 24·46-s − 9·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.894·5-s + 0.755·7-s − 1.26·10-s + 4.22·11-s − 0.554·13-s − 1.06·14-s − 16-s − 1.14·19-s + 1.34·20-s − 5.96·22-s + 2.50·23-s − 3·25-s + 0.784·26-s + 1.13·28-s − 2.41·29-s + 1.43·31-s + 1.23·32-s + 0.676·35-s + 1.31·37-s + 1.62·38-s − 0.312·41-s + 1.37·43-s + 6.33·44-s − 3.53·46-s − 1.31·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{24} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(8613.12\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{24} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.967520583\)
\(L(\frac12)\) \(\approx\) \(2.967520583\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - 2 T + 2 T^{2} + 19 T^{3} + 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 + p T + T^{2} - p^{2} T^{3} - 7 T^{4} - T^{5} + 7 T^{6} - p T^{7} - 7 p^{2} T^{8} - p^{5} T^{9} + p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
5 \( ( 1 - T + 9 T^{2} - 13 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 7 T + 45 T^{2} - 157 T^{3} + 45 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 + 2 T - 16 T^{2} + 30 T^{3} + 10 p T^{4} - 602 T^{5} - 1457 T^{6} - 602 p T^{7} + 10 p^{3} T^{8} + 30 p^{3} T^{9} - 16 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 18 T^{2} - 18 T^{3} + 18 T^{4} + 162 T^{5} + 4399 T^{6} + 162 p T^{7} + 18 p^{2} T^{8} - 18 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 + 5 T - 28 T^{2} - 3 p T^{3} + 997 T^{4} + 268 T^{5} - 22757 T^{6} + 268 p T^{7} + 997 p^{2} T^{8} - 3 p^{4} T^{9} - 28 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 - 6 T + 72 T^{2} - 267 T^{3} + 72 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 + 13 T + 52 T^{2} - 5 T^{3} + 13 p T^{4} + 11986 T^{5} + 91837 T^{6} + 11986 p T^{7} + 13 p^{3} T^{8} - 5 p^{3} T^{9} + 52 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 8 T - 30 T^{2} + 102 T^{3} + 2506 T^{4} - 1202 T^{5} - 93509 T^{6} - 1202 p T^{7} + 2506 p^{2} T^{8} + 102 p^{3} T^{9} - 30 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 8 T - 42 T^{2} + 150 T^{3} + 3322 T^{4} - 1094 T^{5} - 153041 T^{6} - 1094 p T^{7} + 3322 p^{2} T^{8} + 150 p^{3} T^{9} - 42 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 2 T - 14 T^{2} + 482 T^{3} + 32 T^{4} - 3604 T^{5} + 167563 T^{6} - 3604 p T^{7} + 32 p^{2} T^{8} + 482 p^{3} T^{9} - 14 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 9 T - 36 T^{2} + 293 T^{3} + 2601 T^{4} - 4824 T^{5} - 126453 T^{6} - 4824 p T^{7} + 2601 p^{2} T^{8} + 293 p^{3} T^{9} - 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 9 T - 18 T^{2} - 819 T^{3} - 2547 T^{4} + 20772 T^{5} + 299095 T^{6} + 20772 p T^{7} - 2547 p^{2} T^{8} - 819 p^{3} T^{9} - 18 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 24 T + 252 T^{2} + 2202 T^{3} + 20916 T^{4} + 146148 T^{5} + 883411 T^{6} + 146148 p T^{7} + 20916 p^{2} T^{8} + 2202 p^{3} T^{9} + 252 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 15 T - 18 T^{2} + 57 T^{3} + 16947 T^{4} - 71898 T^{5} - 430157 T^{6} - 71898 p T^{7} + 16947 p^{2} T^{8} + 57 p^{3} T^{9} - 18 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - T - 133 T^{2} - 132 T^{3} + 9781 T^{4} + 12493 T^{5} - 645074 T^{6} + 12493 p T^{7} + 9781 p^{2} T^{8} - 132 p^{3} T^{9} - 133 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 14 T - 58 T^{2} - 258 T^{3} + 20800 T^{4} + 70720 T^{5} - 964241 T^{6} + 70720 p T^{7} + 20800 p^{2} T^{8} - 258 p^{3} T^{9} - 58 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 3 T + 105 T^{2} + 669 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 7 T - 36 T^{2} + 513 T^{3} + 733 T^{4} - 46082 T^{5} - 8831 T^{6} - 46082 p T^{7} + 733 p^{2} T^{8} + 513 p^{3} T^{9} - 36 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 6 T - 132 T^{2} - 634 T^{3} + 10026 T^{4} + 16110 T^{5} - 794253 T^{6} + 16110 p T^{7} + 10026 p^{2} T^{8} - 634 p^{3} T^{9} - 132 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 3 T - 60 T^{2} - 2247 T^{3} - 4647 T^{4} + 70968 T^{5} + 2191939 T^{6} + 70968 p T^{7} - 4647 p^{2} T^{8} - 2247 p^{3} T^{9} - 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 5 T - 149 T^{2} - 68 T^{3} + 12785 T^{4} + 45481 T^{5} - 1321850 T^{6} + 45481 p T^{7} + 12785 p^{2} T^{8} - 68 p^{3} T^{9} - 149 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 14 T - 111 T^{2} + 1086 T^{3} + 26782 T^{4} - 124358 T^{5} - 1914107 T^{6} - 124358 p T^{7} + 26782 p^{2} T^{8} + 1086 p^{3} T^{9} - 111 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.88133668714538253411718734391, −5.68677239717314368547769977453, −5.64985043215694704525581640847, −5.24888869374364071778697137155, −5.21456784379117538134893399473, −4.80799066992221110620881414263, −4.53770287125160215708016068487, −4.43235642774415847192785454412, −4.40561516771989128806122782060, −4.21916549435895103407059861326, −4.09638225068210606358676429669, −3.67793124029787508627925722009, −3.62235277381301415818517131117, −3.55629099754998273934435840120, −3.02596881204027081117364815903, −2.83525324233807260717436037536, −2.79085624074829133547276040809, −2.13908428662335863450845224829, −2.07433659389344923663832032849, −1.91423148426445952697347471268, −1.68786013476877439140836394908, −1.37230202456763845998654221197, −1.31306302187037574242251857078, −1.11671105719703351188912630282, −0.41064369656042553662034201201, 0.41064369656042553662034201201, 1.11671105719703351188912630282, 1.31306302187037574242251857078, 1.37230202456763845998654221197, 1.68786013476877439140836394908, 1.91423148426445952697347471268, 2.07433659389344923663832032849, 2.13908428662335863450845224829, 2.79085624074829133547276040809, 2.83525324233807260717436037536, 3.02596881204027081117364815903, 3.55629099754998273934435840120, 3.62235277381301415818517131117, 3.67793124029787508627925722009, 4.09638225068210606358676429669, 4.21916549435895103407059861326, 4.40561516771989128806122782060, 4.43235642774415847192785454412, 4.53770287125160215708016068487, 4.80799066992221110620881414263, 5.21456784379117538134893399473, 5.24888869374364071778697137155, 5.64985043215694704525581640847, 5.68677239717314368547769977453, 5.88133668714538253411718734391

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

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