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Properties

Label 20-712e10-1.1-c0e10-0-0
Degree $20$
Conductor $3.348\times 10^{28}$
Sign $1$
Analytic cond. $3.20901\times 10^{-5}$
Root an. cond. $0.596099$
Motivic weight $0$
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 − 9-s − 9·11-s − 9·17-s + 18-s + 9·22-s − 25-s + 9·34-s + 49-s + 50-s + 2·67-s + 2·73-s − 89-s − 2·97-s − 98-s + 9·99-s − 2·107-s + 45·121-s + 127-s + 131-s − 2·134-s + 137-s + 139-s − 2·146-s + 149-s + 151-s + 9·153-s + ⋯
L(s)  = 1  − 2-s − 9-s − 9·11-s − 9·17-s + 18-s + 9·22-s − 25-s + 9·34-s + 49-s + 50-s + 2·67-s + 2·73-s − 89-s − 2·97-s − 98-s + 9·99-s − 2·107-s + 45·121-s + 127-s + 131-s − 2·134-s + 137-s + 139-s − 2·146-s + 149-s + 151-s + 9·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 89^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 89^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{30} \cdot 89^{10}\)
Sign: $1$
Analytic conductor: \(3.20901\times 10^{-5}\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{30} \cdot 89^{10} ,\ ( \ : [0]^{10} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001348047226\)
\(L(\frac12)\) \(\approx\) \(0.001348047226\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
89 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
good3 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
7 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
11 \( ( 1 + T )^{10}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} ) \)
13 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
17 \( ( 1 + T )^{10}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} ) \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
23 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
29 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
31 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
37 \( ( 1 + T^{2} )^{10} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
61 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.20545005577961408594821184555, −4.16941828939288576220831039761, −4.13466401347863861677729704703, −3.84222412218499517445491517628, −3.55262675489948510711511456741, −3.43157732222262826565994375778, −3.41065865557710097289770786194, −3.33775286241059952292018196704, −2.93206283480255289774486273203, −2.79952127318780285046360097758, −2.75052503911364992345911355660, −2.74995916188776201885232267486, −2.72847182945239707993008912618, −2.46217377107256270752906095520, −2.42977696132648258764099142105, −2.40731526164756565277565590906, −2.30832069941141075554432547656, −2.04279115859931417850408170325, −2.03093095829244569893584446012, −1.80127156064046227071770881849, −1.80061388911622004169962072277, −1.70795619226176437029510960185, −0.70134381610135679971868089363, −0.60594101411000472404777510877, −0.087720807979008825358061568418, 0.087720807979008825358061568418, 0.60594101411000472404777510877, 0.70134381610135679971868089363, 1.70795619226176437029510960185, 1.80061388911622004169962072277, 1.80127156064046227071770881849, 2.03093095829244569893584446012, 2.04279115859931417850408170325, 2.30832069941141075554432547656, 2.40731526164756565277565590906, 2.42977696132648258764099142105, 2.46217377107256270752906095520, 2.72847182945239707993008912618, 2.74995916188776201885232267486, 2.75052503911364992345911355660, 2.79952127318780285046360097758, 2.93206283480255289774486273203, 3.33775286241059952292018196704, 3.41065865557710097289770786194, 3.43157732222262826565994375778, 3.55262675489948510711511456741, 3.84222412218499517445491517628, 4.13466401347863861677729704703, 4.16941828939288576220831039761, 4.20545005577961408594821184555

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

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