Properties

Label 2-108-1.1-c3-0-0
Degree $2$
Conductor $108$
Sign $1$
Analytic cond. $6.37220$
Root an. cond. $2.52432$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 17·7-s + 89·13-s + 107·19-s − 125·25-s + 308·31-s − 433·37-s − 520·43-s − 54·49-s − 901·61-s + 1.00e3·67-s − 271·73-s + 503·79-s + 1.51e3·91-s + 1.85e3·97-s − 19·103-s − 646·109-s + ⋯
L(s)  = 1  + 0.917·7-s + 1.89·13-s + 1.29·19-s − 25-s + 1.78·31-s − 1.92·37-s − 1.84·43-s − 0.157·49-s − 1.89·61-s + 1.83·67-s − 0.434·73-s + 0.716·79-s + 1.74·91-s + 1.93·97-s − 0.0181·103-s − 0.567·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(6.37220\)
Root analytic conductor: \(2.52432\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.809555646\)
\(L(\frac12)\) \(\approx\) \(1.809555646\)
\(L(\frac{5}{2})\) 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 \( 1 + p^{3} T^{2} \)
7 \( 1 - 17 T + p^{3} T^{2} \)
11 \( 1 + p^{3} T^{2} \)
13 \( 1 - 89 T + p^{3} T^{2} \)
17 \( 1 + p^{3} T^{2} \)
19 \( 1 - 107 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + p^{3} T^{2} \)
31 \( 1 - 308 T + p^{3} T^{2} \)
37 \( 1 + 433 T + p^{3} T^{2} \)
41 \( 1 + p^{3} T^{2} \)
43 \( 1 + 520 T + p^{3} T^{2} \)
47 \( 1 + p^{3} T^{2} \)
53 \( 1 + p^{3} T^{2} \)
59 \( 1 + p^{3} T^{2} \)
61 \( 1 + 901 T + p^{3} T^{2} \)
67 \( 1 - 1007 T + p^{3} T^{2} \)
71 \( 1 + p^{3} T^{2} \)
73 \( 1 + 271 T + p^{3} T^{2} \)
79 \( 1 - 503 T + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 + p^{3} T^{2} \)
97 \( 1 - 1853 T + p^{3} T^{2} \)
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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

−13.52502027016020065003359387203, −11.95640490879987010495341009463, −11.24577919624522818887574311912, −10.11730912694271761552726720919, −8.720903511000079700734112084532, −7.87525169380022541520549168122, −6.37419078895437798106800333831, −5.07470900346204015063347868938, −3.54014402364140883660609723275, −1.42760656877177586627104493374, 1.42760656877177586627104493374, 3.54014402364140883660609723275, 5.07470900346204015063347868938, 6.37419078895437798106800333831, 7.87525169380022541520549168122, 8.720903511000079700734112084532, 10.11730912694271761552726720919, 11.24577919624522818887574311912, 11.95640490879987010495341009463, 13.52502027016020065003359387203

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