↑

Properties

Label 18-1607e9-1607.1606-c0e9-0-0
Degree $18$
Conductor $7.147\times 10^{28}$
Sign $1$
Analytic cond. $0.137264$
Root an. cond. $0.895543$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·8-s + 9·25-s + 9·49-s + 3·64-s − 9·73-s + 9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 27·200-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 3·8-s + 9·25-s + 9·49-s + 3·64-s − 9·73-s + 9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 27·200-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1607^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1607^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(1607^{9}\)
Sign: $1$
Analytic conductor: \(0.137264\)
Root analytic conductor: \(0.895543\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1607} (1606, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 1607^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.336935455\)
\(L(\frac12)\) \(\approx\) \(1.336935455\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1607 \( ( 1 - T )^{9} \)
good2 \( ( 1 + T^{3} + T^{6} )^{3} \)
3 \( 1 + T^{9} + T^{18} \)
5 \( ( 1 - T )^{9}( 1 + T )^{9} \)
7 \( ( 1 - T )^{9}( 1 + T )^{9} \)
11 \( ( 1 - T )^{9}( 1 + T )^{9} \)
13 \( ( 1 - T )^{9}( 1 + T )^{9} \)
17 \( 1 + T^{9} + T^{18} \)
19 \( ( 1 - T )^{9}( 1 + T )^{9} \)
23 \( 1 + T^{9} + T^{18} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( 1 + T^{9} + T^{18} \)
37 \( 1 + T^{9} + T^{18} \)
41 \( 1 + T^{9} + T^{18} \)
43 \( ( 1 - T )^{9}( 1 + T )^{9} \)
47 \( ( 1 - T )^{9}( 1 + T )^{9} \)
53 \( 1 + T^{9} + T^{18} \)
59 \( 1 + T^{9} + T^{18} \)
61 \( ( 1 - T )^{9}( 1 + T )^{9} \)
67 \( 1 + T^{9} + T^{18} \)
71 \( ( 1 - T )^{9}( 1 + T )^{9} \)
73 \( ( 1 + T + T^{2} )^{9} \)
79 \( 1 + T^{9} + T^{18} \)
83 \( ( 1 - T )^{9}( 1 + T )^{9} \)
89 \( 1 + T^{9} + T^{18} \)
97 \( ( 1 - T )^{9}( 1 + T )^{9} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.83704681594124540319193283701, −3.67110231156475095129878454569, −3.51343804671154052981143156080, −3.48519098581215604705717760112, −3.41741665918111826187846979660, −3.07401070854036900596088238482, −3.05144859236996630972521605356, −3.03346873858531218537788186781, −2.93510711179110202387640299273, −2.89365410026684000581360346072, −2.88720863773957560086753657428, −2.67360178539299749714820448501, −2.66640141269749633069211881035, −2.35679015386856533845640598345, −2.33625550520959911819304052554, −2.21231238785732949847236678059, −1.93358471655123211921590576680, −1.83165881356381665565982940335, −1.66468087756908046075752713195, −1.37995447452642548139657328305, −1.19631939486700913431876224970, −0.963630090356685194565330324816, −0.863457944470530516935021326909, −0.807959216443588447647770002932, −0.76857505771791767930893930804, 0.76857505771791767930893930804, 0.807959216443588447647770002932, 0.863457944470530516935021326909, 0.963630090356685194565330324816, 1.19631939486700913431876224970, 1.37995447452642548139657328305, 1.66468087756908046075752713195, 1.83165881356381665565982940335, 1.93358471655123211921590576680, 2.21231238785732949847236678059, 2.33625550520959911819304052554, 2.35679015386856533845640598345, 2.66640141269749633069211881035, 2.67360178539299749714820448501, 2.88720863773957560086753657428, 2.89365410026684000581360346072, 2.93510711179110202387640299273, 3.03346873858531218537788186781, 3.05144859236996630972521605356, 3.07401070854036900596088238482, 3.41741665918111826187846979660, 3.48519098581215604705717760112, 3.51343804671154052981143156080, 3.67110231156475095129878454569, 3.83704681594124540319193283701

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

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