Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.700 + 0.713i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.85 − 0.756i)2-s + (2.85 − 2.80i)4-s + 1.08·5-s − 6.01i·7-s + (3.16 − 7.34i)8-s + (2.00 − 0.817i)10-s + 17.7i·11-s + 12.4·13-s + (−4.55 − 11.1i)14-s + (0.291 − 15.9i)16-s − 26.3·17-s − 5.19i·19-s + (3.08 − 3.02i)20-s + (13.4 + 32.8i)22-s + 29.8i·23-s + ⋯
L(s)  = 1  + (0.925 − 0.378i)2-s + (0.713 − 0.700i)4-s + 0.216·5-s − 0.859i·7-s + (0.395 − 0.918i)8-s + (0.200 − 0.0817i)10-s + 1.61i·11-s + 0.955·13-s + (−0.325 − 0.795i)14-s + (0.0182 − 0.999i)16-s − 1.55·17-s − 0.273i·19-s + (0.154 − 0.151i)20-s + (0.609 + 1.49i)22-s + 1.29i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.700 + 0.713i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.700 + 0.713i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.11062 - 0.885547i\)
\(L(\frac12)\)  \(\approx\)  \(2.11062 - 0.885547i\)
\(L(2)\)   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 \( 1 + (-1.85 + 0.756i)T \)
3 \( 1 \)
good5 \( 1 - 1.08T + 25T^{2} \)
7 \( 1 + 6.01iT - 49T^{2} \)
11 \( 1 - 17.7iT - 121T^{2} \)
13 \( 1 - 12.4T + 169T^{2} \)
17 \( 1 + 26.3T + 289T^{2} \)
19 \( 1 + 5.19iT - 361T^{2} \)
23 \( 1 - 29.8iT - 529T^{2} \)
29 \( 1 - 4.32T + 841T^{2} \)
31 \( 1 - 44.8iT - 961T^{2} \)
37 \( 1 + 20.4T + 1.36e3T^{2} \)
41 \( 1 - 59.2T + 1.68e3T^{2} \)
43 \( 1 + 19.1iT - 1.84e3T^{2} \)
47 \( 1 + 41.0iT - 2.20e3T^{2} \)
53 \( 1 + 70.0T + 2.80e3T^{2} \)
59 \( 1 + 28.9iT - 3.48e3T^{2} \)
61 \( 1 - 18.4T + 3.72e3T^{2} \)
67 \( 1 + 94.8iT - 4.48e3T^{2} \)
71 \( 1 + 83.8iT - 5.04e3T^{2} \)
73 \( 1 - 55.8T + 5.32e3T^{2} \)
79 \( 1 - 41.0iT - 6.24e3T^{2} \)
83 \( 1 + 35.4iT - 6.88e3T^{2} \)
89 \( 1 + 26.3T + 7.92e3T^{2} \)
97 \( 1 - 76.3T + 9.40e3T^{2} \)
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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

−13.40665944388821051950095633950, −12.46772156375158257145802730569, −11.27295222025833764379652552458, −10.42000998144019217589117003495, −9.328903328624172675902282436627, −7.40617271683920758325301918328, −6.47030170286060828515316692291, −4.89645823495769108944884924577, −3.80552935146742912679130076094, −1.84583082074051754470907594465, 2.56881973252412563258432380126, 4.12254795745738585339832000858, 5.77992919134299362254660276308, 6.34848310842392049074604568836, 8.137571428118512325669404601399, 8.948018946934104578126878769535, 10.90596942032150664069580008750, 11.53777628373762989960932588043, 12.83903099616182611584175394735, 13.58453040795796605662878550081

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