Properties

Degree 20
Conductor $ 2^{10} \cdot 3^{10} \cdot 59^{10} $
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  − 10·2-s − 3-s + 55·4-s + 10·6-s − 2·7-s − 220·8-s + 2·9-s + 4·11-s − 55·12-s + 20·14-s + 715·16-s − 20·18-s − 6·19-s + 2·21-s − 40·22-s − 8·23-s + 220·24-s + 27·25-s − 5·27-s − 110·28-s − 2.00e3·32-s − 4·33-s + 110·36-s + 60·38-s − 20·42-s + 220·44-s + 80·46-s + ⋯
L(s)  = 1  − 7.07·2-s − 0.577·3-s + 55/2·4-s + 4.08·6-s − 0.755·7-s − 77.7·8-s + 2/3·9-s + 1.20·11-s − 15.8·12-s + 5.34·14-s + 178.·16-s − 4.71·18-s − 1.37·19-s + 0.436·21-s − 8.52·22-s − 1.66·23-s + 44.9·24-s + 27/5·25-s − 0.962·27-s − 20.7·28-s − 353.·32-s − 0.696·33-s + 55/3·36-s + 9.73·38-s − 3.08·42-s + 33.1·44-s + 11.7·46-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 59^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 59^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(20\)
\( N \)  =  \(2^{10} \cdot 3^{10} \cdot 59^{10}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{354} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(20,\ 2^{10} \cdot 3^{10} \cdot 59^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )$
$L(1)$  $\approx$  $0.0181330$
$L(\frac12)$  $\approx$  $0.0181330$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;59\}$, \(F_p(T)\) is a polynomial of degree 20. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 19.
$p$$F_p(T)$
bad2 \( ( 1 + T )^{10} \)
3 \( 1 + T - T^{2} + 2 T^{3} + 2 T^{4} - 14 T^{5} + 2 p T^{6} + 2 p^{2} T^{7} - p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
59 \( 1 - 20 T + 71 T^{2} + 896 T^{3} - 6134 T^{4} + 9960 T^{5} - 6134 p T^{6} + 896 p^{2} T^{7} + 71 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
good5 \( 1 - 27 T^{2} + 79 p T^{4} - 3938 T^{6} + 29068 T^{8} - 165022 T^{10} + 29068 p^{2} T^{12} - 3938 p^{4} T^{14} + 79 p^{7} T^{16} - 27 p^{8} T^{18} + p^{10} T^{20} \)
7 \( ( 1 + T + 16 T^{2} + 17 T^{3} + 179 T^{4} + 148 T^{5} + 179 p T^{6} + 17 p^{2} T^{7} + 16 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \)
11 \( ( 1 - 2 T + 30 T^{2} - 34 T^{3} + 439 T^{4} - 372 T^{5} + 439 p T^{6} - 34 p^{2} T^{7} + 30 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
13 \( 1 - 4 p T^{2} + 88 p T^{4} - 10254 T^{6} - 57459 T^{8} + 2166640 T^{10} - 57459 p^{2} T^{12} - 10254 p^{4} T^{14} + 88 p^{7} T^{16} - 4 p^{9} T^{18} + p^{10} T^{20} \)
17 \( 1 - 108 T^{2} + 5426 T^{4} - 171278 T^{6} + 3938173 T^{8} - 73013740 T^{10} + 3938173 p^{2} T^{12} - 171278 p^{4} T^{14} + 5426 p^{6} T^{16} - 108 p^{8} T^{18} + p^{10} T^{20} \)
19 \( ( 1 + 3 T + 77 T^{2} + 178 T^{3} + 2608 T^{4} + 4670 T^{5} + 2608 p T^{6} + 178 p^{2} T^{7} + 77 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
23 \( ( 1 + 4 T + 29 T^{2} + 188 T^{3} + 1084 T^{4} + 2688 T^{5} + 1084 p T^{6} + 188 p^{2} T^{7} + 29 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
29 \( 1 - 155 T^{2} + 12907 T^{4} - 729682 T^{6} + 30825980 T^{8} - 1008918398 T^{10} + 30825980 p^{2} T^{12} - 729682 p^{4} T^{14} + 12907 p^{6} T^{16} - 155 p^{8} T^{18} + p^{10} T^{20} \)
31 \( 1 - 66 T^{2} + 2027 T^{4} - 68420 T^{6} + 2683060 T^{8} - 85000756 T^{10} + 2683060 p^{2} T^{12} - 68420 p^{4} T^{14} + 2027 p^{6} T^{16} - 66 p^{8} T^{18} + p^{10} T^{20} \)
37 \( 1 - 188 T^{2} + 14592 T^{4} - 563614 T^{6} + 9281381 T^{8} - 55855104 T^{10} + 9281381 p^{2} T^{12} - 563614 p^{4} T^{14} + 14592 p^{6} T^{16} - 188 p^{8} T^{18} + p^{10} T^{20} \)
41 \( 1 - 209 T^{2} + 23428 T^{4} - 1817599 T^{6} + 106670111 T^{8} - 4914350696 T^{10} + 106670111 p^{2} T^{12} - 1817599 p^{4} T^{14} + 23428 p^{6} T^{16} - 209 p^{8} T^{18} + p^{10} T^{20} \)
43 \( 1 - 124 T^{2} + 8986 T^{4} - 558234 T^{6} + 29482965 T^{8} - 1340267732 T^{10} + 29482965 p^{2} T^{12} - 558234 p^{4} T^{14} + 8986 p^{6} T^{16} - 124 p^{8} T^{18} + p^{10} T^{20} \)
47 \( ( 1 + 149 T^{2} + 36 T^{3} + 11260 T^{4} + 2520 T^{5} + 11260 p T^{6} + 36 p^{2} T^{7} + 149 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
53 \( 1 - 355 T^{2} + 60783 T^{4} - 6738486 T^{6} + 540355392 T^{8} - 32753382654 T^{10} + 540355392 p^{2} T^{12} - 6738486 p^{4} T^{14} + 60783 p^{6} T^{16} - 355 p^{8} T^{18} + p^{10} T^{20} \)
61 \( 1 - 226 T^{2} + 28531 T^{4} - 2771220 T^{6} + 216253284 T^{8} - 14141970452 T^{10} + 216253284 p^{2} T^{12} - 2771220 p^{4} T^{14} + 28531 p^{6} T^{16} - 226 p^{8} T^{18} + p^{10} T^{20} \)
67 \( 1 - 226 T^{2} + 24469 T^{4} - 2099544 T^{6} + 180779250 T^{8} - 13661508812 T^{10} + 180779250 p^{2} T^{12} - 2099544 p^{4} T^{14} + 24469 p^{6} T^{16} - 226 p^{8} T^{18} + p^{10} T^{20} \)
71 \( 1 - 404 T^{2} + 85792 T^{4} - 12232858 T^{6} + 1288692677 T^{8} - 103955631344 T^{10} + 1288692677 p^{2} T^{12} - 12232858 p^{4} T^{14} + 85792 p^{6} T^{16} - 404 p^{8} T^{18} + p^{10} T^{20} \)
73 \( 1 - 234 T^{2} + 40625 T^{4} - 4874480 T^{6} + 488701726 T^{8} - 38421788236 T^{10} + 488701726 p^{2} T^{12} - 4874480 p^{4} T^{14} + 40625 p^{6} T^{16} - 234 p^{8} T^{18} + p^{10} T^{20} \)
79 \( ( 1 - 3 T + 192 T^{2} - 299 T^{3} + 24163 T^{4} - 44052 T^{5} + 24163 p T^{6} - 299 p^{2} T^{7} + 192 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
83 \( ( 1 - 6 T + 260 T^{2} - 1794 T^{3} + 33211 T^{4} - 211464 T^{5} + 33211 p T^{6} - 1794 p^{2} T^{7} + 260 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
89 \( ( 1 + 8 T + 201 T^{2} + 220 T^{3} + 15682 T^{4} - 31200 T^{5} + 15682 p T^{6} + 220 p^{2} T^{7} + 201 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
97 \( 1 - 682 T^{2} + 227521 T^{4} - 48709872 T^{6} + 7388315838 T^{8} - 828434696780 T^{10} + 7388315838 p^{2} T^{12} - 48709872 p^{4} T^{14} + 227521 p^{6} T^{16} - 682 p^{8} T^{18} + p^{10} T^{20} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.26681078689955658898526289790, −4.12301538351805922609017673628, −4.11011229424872697636413721934, −3.90927441675529685744337903036, −3.75261485573593655758622436758, −3.54850221766906354572284753009, −3.43316960280639529696700690842, −3.23803952970376798417246378690, −3.23681724461727463686502788979, −2.97943538705507058398495672884, −2.88601089769149407945214798624, −2.75059940896852471278931911046, −2.53914572379603289372895817358, −2.45531379445463472890451824723, −2.28211974081840585064491615231, −2.02671837181188182544339155016, −2.00777769652594529976108254401, −1.78897408888218298073846662620, −1.58219246825616697157073117853, −1.27720154074517642947142155742, −1.18898358190915617803119260905, −1.05572735147348133659319009690, −1.04306865061571909889204037014, −0.44038283552706877865809875496, −0.17678319186235787895191156627, 0.17678319186235787895191156627, 0.44038283552706877865809875496, 1.04306865061571909889204037014, 1.05572735147348133659319009690, 1.18898358190915617803119260905, 1.27720154074517642947142155742, 1.58219246825616697157073117853, 1.78897408888218298073846662620, 2.00777769652594529976108254401, 2.02671837181188182544339155016, 2.28211974081840585064491615231, 2.45531379445463472890451824723, 2.53914572379603289372895817358, 2.75059940896852471278931911046, 2.88601089769149407945214798624, 2.97943538705507058398495672884, 3.23681724461727463686502788979, 3.23803952970376798417246378690, 3.43316960280639529696700690842, 3.54850221766906354572284753009, 3.75261485573593655758622436758, 3.90927441675529685744337903036, 4.11011229424872697636413721934, 4.12301538351805922609017673628, 4.26681078689955658898526289790

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

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