↑

Properties

Label 18-2799e9-311.310-c0e9-0-0
Degree $18$
Conductor $1.054\times 10^{31}$
Sign $1$
Analytic cond. $20.2507$
Root an. cond. $1.18189$
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  + 2-s + 5-s − 7-s + 10-s − 13-s − 14-s − 26-s − 35-s + 47-s + 53-s − 65-s − 67-s − 70-s − 73-s − 79-s + 83-s + 89-s + 91-s + 94-s + 106-s + 107-s − 109-s + 113-s + 9·121-s + 127-s − 130-s + 131-s + ⋯
L(s)  = 1  + 2-s + 5-s − 7-s + 10-s − 13-s − 14-s − 26-s − 35-s + 47-s + 53-s − 65-s − 67-s − 70-s − 73-s − 79-s + 83-s + 89-s + 91-s + 94-s + 106-s + 107-s − 109-s + 113-s + 9·121-s + 127-s − 130-s + 131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 311^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 311^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(3^{18} \cdot 311^{9}\)
Sign: $1$
Analytic conductor: \(20.2507\)
Root analytic conductor: \(1.18189\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2799} (1243, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 3^{18} \cdot 311^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.740193133\)
\(L(\frac12)\) \(\approx\) \(2.740193133\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
311 \( ( 1 + T )^{9} \)
good2 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \)
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 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
11 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( ( 1 - T )^{9}( 1 + T )^{9} \)
37 \( ( 1 - T )^{9}( 1 + T )^{9} \)
41 \( ( 1 - T )^{9}( 1 + T )^{9} \)
43 \( ( 1 - T )^{9}( 1 + T )^{9} \)
47 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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

−3.45658503821244575873910992232, −3.44926027201871295410102082164, −3.43171955754796766568923176453, −3.26722695154455277562054206850, −3.20359654099401850181090435609, −3.04620868481976382437410094173, −3.00974812561993129070016098777, −2.81222806963638917601920611448, −2.65030918474046591762538418445, −2.58767455082267125387180516071, −2.47610652071995919829895391547, −2.47269586658855110813565920787, −2.17080436272762217906264049684, −2.10199089236474341433843049152, −2.06186517036231375781463441364, −1.95653104768263513762003955060, −1.87873012211177702476524312306, −1.61156139332477689193397883173, −1.56406747244312336754276787856, −1.29660472184134268146324268608, −1.25203390718293507654500543526, −0.963771994725269595759257012299, −0.893104449886709772116565647283, −0.54809411910123982933926060048, −0.46392315282101679612197314577, 0.46392315282101679612197314577, 0.54809411910123982933926060048, 0.893104449886709772116565647283, 0.963771994725269595759257012299, 1.25203390718293507654500543526, 1.29660472184134268146324268608, 1.56406747244312336754276787856, 1.61156139332477689193397883173, 1.87873012211177702476524312306, 1.95653104768263513762003955060, 2.06186517036231375781463441364, 2.10199089236474341433843049152, 2.17080436272762217906264049684, 2.47269586658855110813565920787, 2.47610652071995919829895391547, 2.58767455082267125387180516071, 2.65030918474046591762538418445, 2.81222806963638917601920611448, 3.00974812561993129070016098777, 3.04620868481976382437410094173, 3.20359654099401850181090435609, 3.26722695154455277562054206850, 3.43171955754796766568923176453, 3.44926027201871295410102082164, 3.45658503821244575873910992232

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

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