Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·5-s − 7-s + 63·11-s − 28·13-s + 72·17-s + 98·19-s + 126·23-s − 44·25-s − 126·29-s − 259·31-s − 9·35-s + 386·37-s − 450·41-s − 34·43-s − 54·47-s − 342·49-s − 693·53-s + 567·55-s + 180·59-s − 280·61-s − 252·65-s − 586·67-s + 504·71-s + 161·73-s − 63·77-s + 440·79-s + 999·83-s + ⋯
L(s)  = 1  + 0.804·5-s − 0.0539·7-s + 1.72·11-s − 0.597·13-s + 1.02·17-s + 1.18·19-s + 1.14·23-s − 0.351·25-s − 0.806·29-s − 1.50·31-s − 0.0434·35-s + 1.71·37-s − 1.71·41-s − 0.120·43-s − 0.167·47-s − 0.997·49-s − 1.79·53-s + 1.39·55-s + 0.397·59-s − 0.587·61-s − 0.480·65-s − 1.06·67-s + 0.842·71-s + 0.258·73-s − 0.0932·77-s + 0.626·79-s + 1.32·83-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  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(1.90043\)
\(L(\frac12)\)  \(\approx\)  \(1.90043\)
\(L(\frac{5}{2})\)   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 \( 1 - 9 T + p^{3} T^{2} \)
7 \( 1 + T + p^{3} T^{2} \)
11 \( 1 - 63 T + p^{3} T^{2} \)
13 \( 1 + 28 T + p^{3} T^{2} \)
17 \( 1 - 72 T + p^{3} T^{2} \)
19 \( 1 - 98 T + p^{3} T^{2} \)
23 \( 1 - 126 T + p^{3} T^{2} \)
29 \( 1 + 126 T + p^{3} T^{2} \)
31 \( 1 + 259 T + p^{3} T^{2} \)
37 \( 1 - 386 T + p^{3} T^{2} \)
41 \( 1 + 450 T + p^{3} T^{2} \)
43 \( 1 + 34 T + p^{3} T^{2} \)
47 \( 1 + 54 T + p^{3} T^{2} \)
53 \( 1 + 693 T + p^{3} T^{2} \)
59 \( 1 - 180 T + p^{3} T^{2} \)
61 \( 1 + 280 T + p^{3} T^{2} \)
67 \( 1 + 586 T + p^{3} T^{2} \)
71 \( 1 - 504 T + p^{3} T^{2} \)
73 \( 1 - 161 T + p^{3} T^{2} \)
79 \( 1 - 440 T + p^{3} T^{2} \)
83 \( 1 - 999 T + p^{3} T^{2} \)
89 \( 1 - 882 T + p^{3} T^{2} \)
97 \( 1 + 721 T + p^{3} T^{2} \)
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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

−13.28917559913371105151489613547, −12.15288330283312116431885897508, −11.22439837725103309426666104910, −9.685695511502878639308345925707, −9.290702780966852884685169905925, −7.58802248976385018640100288420, −6.39057132125290198952260909842, −5.18406504272717641399117477668, −3.44307730524507891492569170438, −1.48366022759503054930530043772, 1.48366022759503054930530043772, 3.44307730524507891492569170438, 5.18406504272717641399117477668, 6.39057132125290198952260909842, 7.58802248976385018640100288420, 9.290702780966852884685169905925, 9.685695511502878639308345925707, 11.22439837725103309426666104910, 12.15288330283312116431885897508, 13.28917559913371105151489613547

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