Properties

Label 18-1676e9-419.418-c0e9-0-0
Degree $18$
Conductor $1.043\times 10^{29}$
Sign $1$
Analytic cond. $0.200391$
Root an. cond. $0.914567$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·107-s + 9·121-s − 3·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  − 9·107-s + 9·121-s − 3·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 419^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 419^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2^{18} \cdot 419^{9}\)
Sign: $1$
Analytic conductor: \(0.200391\)
Root analytic conductor: \(0.914567\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1676} (837, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 2^{18} \cdot 419^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.215731754\)
\(L(\frac12)\) \(\approx\) \(1.215731754\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
419 \( ( 1 - T )^{9} \)
good3 \( 1 + T^{9} + T^{18} \)
5 \( ( 1 + T^{3} + T^{6} )^{3} \)
7 \( 1 + T^{9} + T^{18} \)
11 \( ( 1 - T )^{9}( 1 + T )^{9} \)
13 \( 1 + T^{9} + T^{18} \)
17 \( ( 1 - T )^{9}( 1 + T )^{9} \)
19 \( ( 1 - T )^{9}( 1 + T )^{9} \)
23 \( 1 + T^{9} + T^{18} \)
29 \( 1 + T^{9} + T^{18} \)
31 \( ( 1 - T )^{9}( 1 + T )^{9} \)
37 \( 1 + T^{9} + T^{18} \)
41 \( 1 + T^{9} + T^{18} \)
43 \( ( 1 + T^{3} + T^{6} )^{3} \)
47 \( 1 + T^{9} + T^{18} \)
53 \( ( 1 - T )^{9}( 1 + T )^{9} \)
59 \( 1 + T^{9} + T^{18} \)
61 \( ( 1 - T )^{9}( 1 + T )^{9} \)
67 \( ( 1 - T )^{9}( 1 + T )^{9} \)
71 \( ( 1 - T )^{9}( 1 + T )^{9} \)
73 \( 1 + T^{9} + T^{18} \)
79 \( 1 + T^{9} + T^{18} \)
83 \( ( 1 - T )^{9}( 1 + T )^{9} \)
89 \( ( 1 - T )^{9}( 1 + T )^{9} \)
97 \( ( 1 + T^{3} + T^{6} )^{3} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.87721824796751087881022593158, −3.67303951443576897549206783202, −3.58527339539763913255401598256, −3.53219083912777896896918187343, −3.37023517317863803666562321540, −3.31574572555050294705032460959, −3.22042173156253992403693613307, −3.19503963367311697645884694499, −2.83255418765032031525044625410, −2.65008713489741765338980161471, −2.60780148605153761526095350510, −2.55764318901993571788248444666, −2.53808315581559930868576070100, −2.47423405537619379461215019819, −2.40652945793503760103474935526, −1.90715463322749938897905488422, −1.88495116373696783686522079848, −1.63562702688464560532315355467, −1.60506382225074005970706581125, −1.53149176527642884379610261094, −1.44195502966376257218302937816, −1.33565728532180028326386106046, −0.72082358672103454948759174130, −0.71646983140580682248183328984, −0.63765598223276998118407857196, 0.63765598223276998118407857196, 0.71646983140580682248183328984, 0.72082358672103454948759174130, 1.33565728532180028326386106046, 1.44195502966376257218302937816, 1.53149176527642884379610261094, 1.60506382225074005970706581125, 1.63562702688464560532315355467, 1.88495116373696783686522079848, 1.90715463322749938897905488422, 2.40652945793503760103474935526, 2.47423405537619379461215019819, 2.53808315581559930868576070100, 2.55764318901993571788248444666, 2.60780148605153761526095350510, 2.65008713489741765338980161471, 2.83255418765032031525044625410, 3.19503963367311697645884694499, 3.22042173156253992403693613307, 3.31574572555050294705032460959, 3.37023517317863803666562321540, 3.53219083912777896896918187343, 3.58527339539763913255401598256, 3.67303951443576897549206783202, 3.87721824796751087881022593158

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

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