Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 57.6·5-s − 188.·7-s − 704.·11-s + 795.·13-s + 180.·17-s − 661.·19-s − 3.63e3·23-s + 196·25-s − 8.02e3·29-s − 3.00e3·31-s − 1.08e4·35-s − 1.52e3·37-s − 3.46e3·41-s + 1.18e4·43-s + 5.97e3·47-s + 1.88e4·49-s − 1.38e4·53-s − 4.05e4·55-s − 2.26e4·59-s − 3.74e4·61-s + 4.58e4·65-s + 7.09e4·67-s + 6.77e4·71-s − 3.15e4·73-s + 1.32e5·77-s + 6.27e4·79-s − 9.34e4·83-s + ⋯
L(s)  = 1  + 1.03·5-s − 1.45·7-s − 1.75·11-s + 1.30·13-s + 0.151·17-s − 0.420·19-s − 1.43·23-s + 0.0627·25-s − 1.77·29-s − 0.561·31-s − 1.50·35-s − 0.182·37-s − 0.321·41-s + 0.974·43-s + 0.394·47-s + 1.12·49-s − 0.675·53-s − 1.80·55-s − 0.846·59-s − 1.28·61-s + 1.34·65-s + 1.93·67-s + 1.59·71-s − 0.693·73-s + 2.55·77-s + 1.13·79-s − 1.48·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(5\)
character  :  $\chi_{108} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -1)\)
\(L(3)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{7}{2})\)   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 \( 1 - 57.6T + 3.12e3T^{2} \)
7 \( 1 + 188.T + 1.68e4T^{2} \)
11 \( 1 + 704.T + 1.61e5T^{2} \)
13 \( 1 - 795.T + 3.71e5T^{2} \)
17 \( 1 - 180.T + 1.41e6T^{2} \)
19 \( 1 + 661.T + 2.47e6T^{2} \)
23 \( 1 + 3.63e3T + 6.43e6T^{2} \)
29 \( 1 + 8.02e3T + 2.05e7T^{2} \)
31 \( 1 + 3.00e3T + 2.86e7T^{2} \)
37 \( 1 + 1.52e3T + 6.93e7T^{2} \)
41 \( 1 + 3.46e3T + 1.15e8T^{2} \)
43 \( 1 - 1.18e4T + 1.47e8T^{2} \)
47 \( 1 - 5.97e3T + 2.29e8T^{2} \)
53 \( 1 + 1.38e4T + 4.18e8T^{2} \)
59 \( 1 + 2.26e4T + 7.14e8T^{2} \)
61 \( 1 + 3.74e4T + 8.44e8T^{2} \)
67 \( 1 - 7.09e4T + 1.35e9T^{2} \)
71 \( 1 - 6.77e4T + 1.80e9T^{2} \)
73 \( 1 + 3.15e4T + 2.07e9T^{2} \)
79 \( 1 - 6.27e4T + 3.07e9T^{2} \)
83 \( 1 + 9.34e4T + 3.93e9T^{2} \)
89 \( 1 - 6.87e4T + 5.58e9T^{2} \)
97 \( 1 - 1.17e5T + 8.58e9T^{2} \)
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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

−12.66321745753485678525558099401, −10.89412472095523456153149960272, −10.07402872452429506893550481858, −9.199154689048109096497305727703, −7.79586544670057594365706128324, −6.25936863139840133919971146252, −5.59754178581492614688608002415, −3.56943105237776466554535272994, −2.15948891911118990100055856966, 0, 2.15948891911118990100055856966, 3.56943105237776466554535272994, 5.59754178581492614688608002415, 6.25936863139840133919971146252, 7.79586544670057594365706128324, 9.199154689048109096497305727703, 10.07402872452429506893550481858, 10.89412472095523456153149960272, 12.66321745753485678525558099401

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