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Properties

Label 2-108-3.2-c0-0-0
Degree $2$
Conductor $108$
Sign $1$
Analytic cond. $0.0538990$
Root an. cond. $0.232161$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 13-s − 19-s + 25-s + 2·31-s − 37-s + 2·43-s − 61-s − 67-s − 73-s − 79-s + 91-s − 97-s − 103-s + 2·109-s + ⋯
L(s)  = 1  − 7-s − 13-s − 19-s + 25-s + 2·31-s − 37-s + 2·43-s − 61-s − 67-s − 73-s − 79-s + 91-s − 97-s − 103-s + 2·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(0.0538990\)
Root analytic conductor: \(0.232161\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{108} (53, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5402022062\)
\(L(\frac12)\) \(\approx\) \(0.5402022062\)
\(L(1)\) 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 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )^{2} \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.92836271074950410633497392280, −12.81147910151262800003561171965, −12.11393391520275622911136162744, −10.66337417717924861137567235324, −9.764363290770792093036581350306, −8.645382261510086690609111542994, −7.20655230074699554971488628053, −6.14349374284521055741894837740, −4.54012115448883134462155601253, −2.82161410923512167788607608967, 2.82161410923512167788607608967, 4.54012115448883134462155601253, 6.14349374284521055741894837740, 7.20655230074699554971488628053, 8.645382261510086690609111542994, 9.764363290770792093036581350306, 10.66337417717924861137567235324, 12.11393391520275622911136162744, 12.81147910151262800003561171965, 13.92836271074950410633497392280

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