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Properties

Label 12-3381e6-1.1-c1e6-0-1
Degree $12$
Conductor $1.494\times 10^{21}$
Sign $1$
Analytic cond. $3.87198\times 10^{8}$
Root an. cond. $5.19590$
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  − 2-s − 6·3-s − 4·4-s + 3·5-s + 6·6-s + 4·8-s + 21·9-s − 3·10-s − 14·11-s + 24·12-s − 18·15-s + 5·16-s + 15·17-s − 21·18-s − 19-s − 12·20-s + 14·22-s − 6·23-s − 24·24-s − 6·25-s − 56·27-s − 6·29-s + 18·30-s − 11·31-s − 3·32-s + 84·33-s − 15·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 3.46·3-s − 2·4-s + 1.34·5-s + 2.44·6-s + 1.41·8-s + 7·9-s − 0.948·10-s − 4.22·11-s + 6.92·12-s − 4.64·15-s + 5/4·16-s + 3.63·17-s − 4.94·18-s − 0.229·19-s − 2.68·20-s + 2.98·22-s − 1.25·23-s − 4.89·24-s − 6/5·25-s − 10.7·27-s − 1.11·29-s + 3.28·30-s − 1.97·31-s − 0.530·32-s + 14.6·33-s − 2.57·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 7^{12} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(3.87198\times 10^{8}\)
Root analytic conductor: \(5.19590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{6} \cdot 7^{12} \cdot 23^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad3 \( ( 1 + T )^{6} \)
7 \( 1 \)
23 \( ( 1 + T )^{6} \)
good2 \( 1 + T + 5 T^{2} + 5 T^{3} + p^{4} T^{4} + 7 p T^{5} + 19 p T^{6} + 7 p^{2} T^{7} + p^{6} T^{8} + 5 p^{3} T^{9} + 5 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 3 T + 3 p T^{2} - 8 p T^{3} + 127 T^{4} - 59 p T^{5} + 696 T^{6} - 59 p^{2} T^{7} + 127 p^{2} T^{8} - 8 p^{4} T^{9} + 3 p^{5} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 14 T + 118 T^{2} + 746 T^{3} + 3783 T^{4} + 16020 T^{5} + 57588 T^{6} + 16020 p T^{7} + 3783 p^{2} T^{8} + 746 p^{3} T^{9} + 118 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 47 T^{2} + 4 p T^{3} + 970 T^{4} + 2024 T^{5} + 13727 T^{6} + 2024 p T^{7} + 970 p^{2} T^{8} + 4 p^{4} T^{9} + 47 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 15 T + 163 T^{2} - 1298 T^{3} + 8385 T^{4} - 44433 T^{5} + 200684 T^{6} - 44433 p T^{7} + 8385 p^{2} T^{8} - 1298 p^{3} T^{9} + 163 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + T + 49 T^{2} - 62 T^{3} + 549 T^{4} - 277 p T^{5} - 802 T^{6} - 277 p^{2} T^{7} + 549 p^{2} T^{8} - 62 p^{3} T^{9} + 49 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 6 T + 108 T^{2} + 456 T^{3} + 5671 T^{4} + 18570 T^{5} + 191272 T^{6} + 18570 p T^{7} + 5671 p^{2} T^{8} + 456 p^{3} T^{9} + 108 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 11 T + 187 T^{2} + 1562 T^{3} + 14733 T^{4} + 92423 T^{5} + 613542 T^{6} + 92423 p T^{7} + 14733 p^{2} T^{8} + 1562 p^{3} T^{9} + 187 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 5 T + 183 T^{2} + 22 p T^{3} + 15033 T^{4} + 56893 T^{5} + 713118 T^{6} + 56893 p T^{7} + 15033 p^{2} T^{8} + 22 p^{4} T^{9} + 183 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 18 T + 220 T^{2} - 1492 T^{3} + 8047 T^{4} - 22018 T^{5} + 110984 T^{6} - 22018 p T^{7} + 8047 p^{2} T^{8} - 1492 p^{3} T^{9} + 220 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 37 T + 761 T^{2} + 10914 T^{3} + 120293 T^{4} + 1061333 T^{5} + 7670238 T^{6} + 1061333 p T^{7} + 120293 p^{2} T^{8} + 10914 p^{3} T^{9} + 761 p^{4} T^{10} + 37 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 3 T + 169 T^{2} - 526 T^{3} + 13951 T^{4} - 41799 T^{5} + 767630 T^{6} - 41799 p T^{7} + 13951 p^{2} T^{8} - 526 p^{3} T^{9} + 169 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 15 T + 227 T^{2} + 1518 T^{3} + 12301 T^{4} + 37347 T^{5} + 400302 T^{6} + 37347 p T^{7} + 12301 p^{2} T^{8} + 1518 p^{3} T^{9} + 227 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 2 T + 152 T^{2} - 4 T^{3} + 13279 T^{4} - 12750 T^{5} + 843264 T^{6} - 12750 p T^{7} + 13279 p^{2} T^{8} - 4 p^{3} T^{9} + 152 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 12 T + 202 T^{2} + 1516 T^{3} + 15911 T^{4} + 105816 T^{5} + 991468 T^{6} + 105816 p T^{7} + 15911 p^{2} T^{8} + 1516 p^{3} T^{9} + 202 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 10 T + 327 T^{2} + 2466 T^{3} + 47774 T^{4} + 289718 T^{5} + 4101235 T^{6} + 289718 p T^{7} + 47774 p^{2} T^{8} + 2466 p^{3} T^{9} + 327 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 21 T + 503 T^{2} + 6682 T^{3} + 92121 T^{4} + 885629 T^{5} + 8722550 T^{6} + 885629 p T^{7} + 92121 p^{2} T^{8} + 6682 p^{3} T^{9} + 503 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 8 T + 223 T^{2} + 1356 T^{3} + 28298 T^{4} + 157856 T^{5} + 2588167 T^{6} + 157856 p T^{7} + 28298 p^{2} T^{8} + 1356 p^{3} T^{9} + 223 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 17 T + 175 T^{2} + 816 T^{3} + 1705 T^{4} - 87841 T^{5} - 1049430 T^{6} - 87841 p T^{7} + 1705 p^{2} T^{8} + 816 p^{3} T^{9} + 175 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 12 T + 418 T^{2} - 3758 T^{3} + 75163 T^{4} - 531066 T^{5} + 7859204 T^{6} - 531066 p T^{7} + 75163 p^{2} T^{8} - 3758 p^{3} T^{9} + 418 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 18 T + 506 T^{2} - 7324 T^{3} + 107607 T^{4} - 1244842 T^{5} + 12553156 T^{6} - 1244842 p T^{7} + 107607 p^{2} T^{8} - 7324 p^{3} T^{9} + 506 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 2 T + 354 T^{2} + 290 T^{3} + 58303 T^{4} - 14692 T^{5} + 6475708 T^{6} - 14692 p T^{7} + 58303 p^{2} T^{8} + 290 p^{3} T^{9} + 354 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.03280564333685543035445025294, −4.89617029917841762770179652193, −4.70936889141501132749456366423, −4.49967637567644292453632006168, −4.46877088050710787955109288420, −4.36139186096881899251811901785, −4.20362203903959719944076108976, −3.74804253119229425166022467574, −3.70825862883724760151154719196, −3.66200637203610797145064834959, −3.50818803575482742990229973466, −3.50073678803794701649485819939, −3.08637088953727750778795066805, −2.95663934279639995700306735885, −2.73354199150103204961218534721, −2.60897179801659723337781183865, −2.46037526772057143821177388677, −2.04086540020696863110292642239, −2.02773190174070506546621259357, −1.90329395284133856898625048937, −1.41161963702415896113339356655, −1.39980444621589590491967583280, −1.37982863221226548384402762977, −1.14868214418247025152375760072, −0.881795175590712851918125535423, 0, 0, 0, 0, 0, 0, 0.881795175590712851918125535423, 1.14868214418247025152375760072, 1.37982863221226548384402762977, 1.39980444621589590491967583280, 1.41161963702415896113339356655, 1.90329395284133856898625048937, 2.02773190174070506546621259357, 2.04086540020696863110292642239, 2.46037526772057143821177388677, 2.60897179801659723337781183865, 2.73354199150103204961218534721, 2.95663934279639995700306735885, 3.08637088953727750778795066805, 3.50073678803794701649485819939, 3.50818803575482742990229973466, 3.66200637203610797145064834959, 3.70825862883724760151154719196, 3.74804253119229425166022467574, 4.20362203903959719944076108976, 4.36139186096881899251811901785, 4.46877088050710787955109288420, 4.49967637567644292453632006168, 4.70936889141501132749456366423, 4.89617029917841762770179652193, 5.03280564333685543035445025294

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

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