Properties

Degree $12$
Conductor $62523502209$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4·3-s + 2·4-s + 5·5-s − 4·6-s + 3·7-s − 8-s + 6·9-s + 5·10-s + 2·11-s − 8·12-s − 3·13-s + 3·14-s − 20·15-s − 24·17-s + 6·18-s − 6·19-s + 10·20-s − 12·21-s + 2·22-s + 4·24-s + 17·25-s − 3·26-s − 5·27-s + 6·28-s − 29-s − 20·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 2.30·3-s + 4-s + 2.23·5-s − 1.63·6-s + 1.13·7-s − 0.353·8-s + 2·9-s + 1.58·10-s + 0.603·11-s − 2.30·12-s − 0.832·13-s + 0.801·14-s − 5.16·15-s − 5.82·17-s + 1.41·18-s − 1.37·19-s + 2.23·20-s − 2.61·21-s + 0.426·22-s + 0.816·24-s + 17/5·25-s − 0.588·26-s − 0.962·27-s + 1.13·28-s − 0.185·29-s − 3.65·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{6}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{63} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.504261\)
\(L(\frac12)\) \(\approx\) \(0.504261\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 4 T + 10 T^{2} + 7 p T^{3} + 10 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
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

−8.959962920203876046251110770482, −8.533101499363056205200552398851, −8.192470646892910787585094736683, −7.86115365477267947192291609274, −7.81739185740669098098179593972, −7.16857274627485590499394876326, −7.02949503817024933281254373927, −6.63714603147415688667459483870, −6.58896046172724283742157308827, −6.52446111003679327033583646723, −6.27440148941157616623710342386, −6.18579846603422312459981074019, −5.76824547598489157083295571479, −5.53236216195131829681392088431, −5.52690236278043528756984142182, −5.01523058526717656416219994446, −4.70683705393440573712557272699, −4.51715697813269216610725851510, −4.37226227371296963193459256221, −4.25055572786579143279363424997, −3.40962639127877897477938134139, −2.77665001478225031152817436608, −2.25819598315759132741083676837, −2.09213707624128707708651459098, −2.07543646350802292275723653096, 2.07543646350802292275723653096, 2.09213707624128707708651459098, 2.25819598315759132741083676837, 2.77665001478225031152817436608, 3.40962639127877897477938134139, 4.25055572786579143279363424997, 4.37226227371296963193459256221, 4.51715697813269216610725851510, 4.70683705393440573712557272699, 5.01523058526717656416219994446, 5.52690236278043528756984142182, 5.53236216195131829681392088431, 5.76824547598489157083295571479, 6.18579846603422312459981074019, 6.27440148941157616623710342386, 6.52446111003679327033583646723, 6.58896046172724283742157308827, 6.63714603147415688667459483870, 7.02949503817024933281254373927, 7.16857274627485590499394876326, 7.81739185740669098098179593972, 7.86115365477267947192291609274, 8.192470646892910787585094736683, 8.533101499363056205200552398851, 8.959962920203876046251110770482

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

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