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Properties

Label 12-189e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.558\times 10^{13}$
Sign $1$
Analytic cond. $11.8149$
Root an. cond. $1.22848$
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-s + 2·4-s − 5·5-s + 3·7-s + 8-s + 5·10-s − 2·11-s − 3·13-s − 3·14-s + 24·17-s − 6·19-s − 10·20-s + 2·22-s + 17·25-s + 3·26-s + 6·28-s + 29-s + 3·31-s + 4·32-s − 24·34-s − 15·35-s − 6·37-s + 6·38-s − 5·40-s − 22·41-s + 3·43-s − 4·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 4-s − 2.23·5-s + 1.13·7-s + 0.353·8-s + 1.58·10-s − 0.603·11-s − 0.832·13-s − 0.801·14-s + 5.82·17-s − 1.37·19-s − 2.23·20-s + 0.426·22-s + 17/5·25-s + 0.588·26-s + 1.13·28-s + 0.185·29-s + 0.538·31-s + 0.707·32-s − 4.11·34-s − 2.53·35-s − 0.986·37-s + 0.973·38-s − 0.790·40-s − 3.43·41-s + 0.457·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(11.8149\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.215868317\)
\(L(\frac12)\) \(\approx\) \(1.215868317\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T + T^{2} )^{3} \)
good2 \( 1 + T - T^{2} - p^{2} T^{3} - 3 T^{4} + p T^{5} + 13 T^{6} + p^{2} T^{7} - 3 p^{2} T^{8} - p^{5} T^{9} - p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + p T + 8 T^{2} + 7 T^{3} + 9 T^{4} - 62 T^{5} - 299 T^{6} - 62 p T^{7} + 9 p^{2} T^{8} + 7 p^{3} T^{9} + 8 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T - 10 T^{2} + 34 T^{3} + 48 T^{4} - 416 T^{5} + 31 T^{6} - 416 p T^{7} + 48 p^{2} T^{8} + 34 p^{3} T^{9} - 10 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} )^{3} \)
17 \( ( 1 - 12 T + 90 T^{2} - 435 T^{3} + 90 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 + 3 T + 51 T^{2} + 107 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 36 T^{2} + 18 T^{3} + 468 T^{4} - 324 T^{5} - 5393 T^{6} - 324 p T^{7} + 468 p^{2} T^{8} + 18 p^{3} T^{9} - 36 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 - T - 82 T^{2} + 31 T^{3} + 4425 T^{4} - 758 T^{5} - 148595 T^{6} - 758 p T^{7} + 4425 p^{2} T^{8} + 31 p^{3} T^{9} - 82 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 3 T - 60 T^{2} + 219 T^{3} + 1983 T^{4} - 4746 T^{5} - 51289 T^{6} - 4746 p T^{7} + 1983 p^{2} T^{8} + 219 p^{3} T^{9} - 60 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 3 T + 57 T^{2} + 303 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 22 T + 206 T^{2} + 1802 T^{3} + 18432 T^{4} + 135116 T^{5} + 808243 T^{6} + 135116 p T^{7} + 18432 p^{2} T^{8} + 1802 p^{3} T^{9} + 206 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 13170 p T^{7} + 123 p^{2} T^{8} + 569 p^{3} T^{9} - 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 9 T - 6 T^{2} - 531 T^{3} - 2433 T^{4} + 3438 T^{5} + 104623 T^{6} + 3438 p T^{7} - 2433 p^{2} T^{8} - 531 p^{3} T^{9} - 6 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 18 T + 234 T^{2} - 1917 T^{3} + 234 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 9 T - 90 T^{2} - 459 T^{3} + 10161 T^{4} + 20556 T^{5} - 598421 T^{6} + 20556 p T^{7} + 10161 p^{2} T^{8} - 459 p^{3} T^{9} - 90 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T - 126 T^{2} + 358 T^{3} + 12372 T^{4} - 11472 T^{5} - 838653 T^{6} - 11472 p T^{7} + 12372 p^{2} T^{8} + 358 p^{3} T^{9} - 126 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T^{2} + 1366 T^{3} + 438 T^{4} + 4098 T^{5} + 1065603 T^{6} + 4098 p T^{7} + 438 p^{2} T^{8} + 1366 p^{3} T^{9} + 6 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 + 9 T + 207 T^{2} + 1197 T^{3} + 207 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 3 T + 51 T^{2} - 681 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 + 15 T + 36 T^{2} - 367 T^{3} - 3225 T^{4} - 51726 T^{5} - 676905 T^{6} - 51726 p T^{7} - 3225 p^{2} T^{8} - 367 p^{3} T^{9} + 36 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 12 T - 144 T^{2} - 582 T^{3} + 34812 T^{4} + 90444 T^{5} - 2656433 T^{6} + 90444 p T^{7} + 34812 p^{2} T^{8} - 582 p^{3} T^{9} - 144 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 - 2 T + 116 T^{2} - 735 T^{3} + 116 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 + 3 T - 168 T^{2} + 573 T^{3} + 14223 T^{4} - 78504 T^{5} - 1297807 T^{6} - 78504 p T^{7} + 14223 p^{2} T^{8} + 573 p^{3} T^{9} - 168 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−7.24857727456283483338327968032, −6.77926427235722744338350715869, −6.67953172013841081901892474199, −6.61596037103994374500080168714, −6.43407349601293785210187713794, −5.74539050755666355124291412163, −5.70230577990157987499611332924, −5.40355588632900731165398578474, −5.32023169211293971085883323480, −5.28305784153718558869712343548, −5.19997247781623912807733440146, −4.51028781951976494243216193422, −4.42872773456760787609298956683, −4.29029878668166145768455608950, −4.12623441675620218959095665002, −3.71026619993020913154436508201, −3.30931639574839657528727667067, −3.22831514581994555807523278061, −3.16314228315911781210588835565, −2.90087674200382295163118206298, −2.36787573357677827298945058940, −1.92785582544017823996418248887, −1.41710530139108223374200000506, −1.37079395849934071523478881278, −0.72305490071213197993133300463, 0.72305490071213197993133300463, 1.37079395849934071523478881278, 1.41710530139108223374200000506, 1.92785582544017823996418248887, 2.36787573357677827298945058940, 2.90087674200382295163118206298, 3.16314228315911781210588835565, 3.22831514581994555807523278061, 3.30931639574839657528727667067, 3.71026619993020913154436508201, 4.12623441675620218959095665002, 4.29029878668166145768455608950, 4.42872773456760787609298956683, 4.51028781951976494243216193422, 5.19997247781623912807733440146, 5.28305784153718558869712343548, 5.32023169211293971085883323480, 5.40355588632900731165398578474, 5.70230577990157987499611332924, 5.74539050755666355124291412163, 6.43407349601293785210187713794, 6.61596037103994374500080168714, 6.67953172013841081901892474199, 6.77926427235722744338350715869, 7.24857727456283483338327968032

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

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