Properties

Label 2-33e2-1.1-c1-0-7
Degree $2$
Conductor $1089$
Sign $1$
Analytic cond. $8.69570$
Root an. cond. $2.94884$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 4·5-s + 2·7-s − 3·8-s − 4·10-s + 2·13-s + 2·14-s − 16-s − 2·17-s + 6·19-s + 4·20-s + 4·23-s + 11·25-s + 2·26-s − 2·28-s + 6·29-s + 4·31-s + 5·32-s − 2·34-s − 8·35-s − 6·37-s + 6·38-s + 12·40-s + 10·41-s − 6·43-s + 4·46-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.78·5-s + 0.755·7-s − 1.06·8-s − 1.26·10-s + 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s + 1.37·19-s + 0.894·20-s + 0.834·23-s + 11/5·25-s + 0.392·26-s − 0.377·28-s + 1.11·29-s + 0.718·31-s + 0.883·32-s − 0.342·34-s − 1.35·35-s − 0.986·37-s + 0.973·38-s + 1.89·40-s + 1.56·41-s − 0.914·43-s + 0.589·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(8.69570\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1089} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.424956829\)
\(L(\frac12)\) \(\approx\) \(1.424956829\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.851570389577448581336707038653, −8.702375538631797576207794069914, −8.310039341701636953836538691364, −7.45345434128145669094314403830, −6.51037497293636656959710624288, −5.14146744035381483907036545967, −4.65639191028282856260150309872, −3.75599253613222283098497330601, −3.03530616104555310081477111018, −0.843916200143676929117242957092, 0.843916200143676929117242957092, 3.03530616104555310081477111018, 3.75599253613222283098497330601, 4.65639191028282856260150309872, 5.14146744035381483907036545967, 6.51037497293636656959710624288, 7.45345434128145669094314403830, 8.310039341701636953836538691364, 8.702375538631797576207794069914, 9.851570389577448581336707038653

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