Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 37·7-s − 19·13-s − 163·19-s − 125·25-s + 308·31-s + 323·37-s − 520·43-s + 1.02e3·49-s + 719·61-s − 127·67-s − 919·73-s − 1.38e3·79-s + 703·91-s − 523·97-s − 1.80e3·103-s − 646·109-s + ⋯
L(s)  = 1  − 1.99·7-s − 0.405·13-s − 1.96·19-s − 25-s + 1.78·31-s + 1.43·37-s − 1.84·43-s + 2.99·49-s + 1.50·61-s − 0.231·67-s − 1.47·73-s − 1.97·79-s + 0.809·91-s − 0.547·97-s − 1.72·103-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

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(3\)
character  :  $\chi_{108} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -1)\)
\(L(2)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + p^{3} T^{2} \)
7 \( 1 + 37 T + p^{3} T^{2} \)
11 \( 1 + p^{3} T^{2} \)
13 \( 1 + 19 T + p^{3} T^{2} \)
17 \( 1 + p^{3} T^{2} \)
19 \( 1 + 163 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + p^{3} T^{2} \)
31 \( 1 - 308 T + p^{3} T^{2} \)
37 \( 1 - 323 T + p^{3} T^{2} \)
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 \( 1 - 719 T + p^{3} T^{2} \)
67 \( 1 + 127 T + p^{3} T^{2} \)
71 \( 1 + p^{3} T^{2} \)
73 \( 1 + 919 T + p^{3} T^{2} \)
79 \( 1 + 1387 T + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 + p^{3} T^{2} \)
97 \( 1 + 523 T + p^{3} T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.86813110321959253387127591130, −11.81662014694048874974144999152, −10.29487780678826224561663286717, −9.659605363844428103309562395955, −8.396127786596266983197280986868, −6.81765522105100052143272051507, −6.05251640242681356935218892129, −4.14720765790314535805919714519, −2.70171297139772601612613454647, 0, 2.70171297139772601612613454647, 4.14720765790314535805919714519, 6.05251640242681356935218892129, 6.81765522105100052143272051507, 8.396127786596266983197280986868, 9.659605363844428103309562395955, 10.29487780678826224561663286717, 11.81662014694048874974144999152, 12.86813110321959253387127591130

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