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Properties

Label 18-2591e9-2591.2590-c0e9-0-0
Degree $18$
Conductor $5.263\times 10^{30}$
Sign $1$
Analytic cond. $10.1069$
Root an. cond. $1.13713$
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  − 2-s − 3-s − 5-s + 6-s + 10-s − 11-s − 13-s + 15-s + 22-s − 23-s + 26-s − 30-s + 33-s − 37-s + 39-s − 41-s + 46-s + 9·49-s − 53-s + 55-s − 59-s + 65-s − 66-s − 67-s + 69-s − 71-s − 73-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s + 10-s − 11-s − 13-s + 15-s + 22-s − 23-s + 26-s − 30-s + 33-s − 37-s + 39-s − 41-s + 46-s + 9·49-s − 53-s + 55-s − 59-s + 65-s − 66-s − 67-s + 69-s − 71-s − 73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2591^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2591^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2591^{9}\)
Sign: $1$
Analytic conductor: \(10.1069\)
Root analytic conductor: \(1.13713\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2591} (2590, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 2591^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1882948161\)
\(L(\frac12)\) \(\approx\) \(0.1882948161\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2591 \( 1+O(T) \)
good2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
3 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
5 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
7 \( ( 1 - T )^{9}( 1 + T )^{9} \)
11 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
13 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
17 \( ( 1 - T )^{9}( 1 + T )^{9} \)
19 \( ( 1 - T )^{9}( 1 + T )^{9} \)
23 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( ( 1 - T )^{9}( 1 + T )^{9} \)
37 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
41 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
43 \( ( 1 - T )^{9}( 1 + T )^{9} \)
47 \( ( 1 - T )^{9}( 1 + T )^{9} \)
53 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
59 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
61 \( ( 1 - T )^{9}( 1 + T )^{9} \)
67 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
71 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
73 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
79 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
83 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
89 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
97 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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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

−3.68208508789130880788584224622, −3.66792925939953768853547203704, −3.40760688542766287134137956122, −3.20148426073223163021810652681, −3.00577223919690535613288490404, −2.90300447029850006388359885370, −2.90257081897064750056196126015, −2.79272785647361970580512956157, −2.74593227905136891784373318362, −2.74249687248141807944029000880, −2.59590265799449571678425252564, −2.55610308149269371580277732894, −2.04971318537310305044173535491, −2.04367221756778186458967405712, −2.02450883192209841945473321420, −1.92671698918816771981720495927, −1.60797197178907916383802461115, −1.60486431422966660633861584212, −1.59553624996496063145931549682, −1.44932157834433651265518281543, −0.890121081650553347220444109154, −0.862114697033660971133337592794, −0.75498071206206051361480389439, −0.45490977865862635648327002336, −0.38089873058529106783027735432, 0.38089873058529106783027735432, 0.45490977865862635648327002336, 0.75498071206206051361480389439, 0.862114697033660971133337592794, 0.890121081650553347220444109154, 1.44932157834433651265518281543, 1.59553624996496063145931549682, 1.60486431422966660633861584212, 1.60797197178907916383802461115, 1.92671698918816771981720495927, 2.02450883192209841945473321420, 2.04367221756778186458967405712, 2.04971318537310305044173535491, 2.55610308149269371580277732894, 2.59590265799449571678425252564, 2.74249687248141807944029000880, 2.74593227905136891784373318362, 2.79272785647361970580512956157, 2.90257081897064750056196126015, 2.90300447029850006388359885370, 3.00577223919690535613288490404, 3.20148426073223163021810652681, 3.40760688542766287134137956122, 3.66792925939953768853547203704, 3.68208508789130880788584224622

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

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