Properties

Label 2-108-1.1-c1-0-0
Degree $2$
Conductor $108$
Sign $1$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
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  + 5·7-s − 7·13-s − 19-s − 5·25-s − 4·31-s − 37-s + 8·43-s + 18·49-s − 13·61-s + 11·67-s + 17·73-s − 13·79-s − 35·91-s + 5·97-s − 7·103-s + 2·109-s + ⋯
L(s)  = 1  + 1.88·7-s − 1.94·13-s − 0.229·19-s − 25-s − 0.718·31-s − 0.164·37-s + 1.21·43-s + 18/7·49-s − 1.66·61-s + 1.34·67-s + 1.98·73-s − 1.46·79-s − 3.66·91-s + 0.507·97-s − 0.689·103-s + 0.191·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{108} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.112912674\)
\(L(\frac12)\) \(\approx\) \(1.112912674\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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

−14.00533773800539081851320941315, −12.48081907715316673289028111467, −11.64280365407061309217144113778, −10.67338269741490887569061453351, −9.420824192029041109275264269235, −8.079851394884847745055740317792, −7.31346310196346190295346056081, −5.41619595580815395232018816358, −4.44194996130619826314058940000, −2.12496210928289498867796346468, 2.12496210928289498867796346468, 4.44194996130619826314058940000, 5.41619595580815395232018816358, 7.31346310196346190295346056081, 8.079851394884847745055740317792, 9.420824192029041109275264269235, 10.67338269741490887569061453351, 11.64280365407061309217144113778, 12.48081907715316673289028111467, 14.00533773800539081851320941315

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