Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.642 - 0.766i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + (2 + 3.46i)5-s + (−3 − 1.73i)7-s − 8·8-s + (−4 − 6.92i)10-s + (10.5 + 6.06i)11-s + (11 + 19.0i)13-s + (6 + 3.46i)14-s + 16·16-s + 11·17-s + 15.5i·19-s + (8 + 13.8i)20-s + (−21 − 12.1i)22-s + (−21 + 12.1i)23-s + ⋯
L(s)  = 1  − 2-s + 4-s + (0.400 + 0.692i)5-s + (−0.428 − 0.247i)7-s − 8-s + (−0.400 − 0.692i)10-s + (0.954 + 0.551i)11-s + (0.846 + 1.46i)13-s + (0.428 + 0.247i)14-s + 16-s + 0.647·17-s + 0.820i·19-s + (0.400 + 0.692i)20-s + (−0.954 − 0.551i)22-s + (−0.913 + 0.527i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.642 - 0.766i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.642 - 0.766i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.849166 + 0.395972i\)
\(L(\frac12)\)  \(\approx\)  \(0.849166 + 0.395972i\)
\(L(2)\)   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 \( 1 + 2T \)
3 \( 1 \)
good5 \( 1 + (-2 - 3.46i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (3 + 1.73i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-10.5 - 6.06i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-11 - 19.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 11T + 289T^{2} \)
19 \( 1 - 15.5iT - 361T^{2} \)
23 \( 1 + (21 - 12.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-17 + 29.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-6 + 3.46i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 16T + 1.36e3T^{2} \)
41 \( 1 + (-6.5 - 11.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (43.5 + 25.1i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (3 + 1.73i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 52T + 2.80e3T^{2} \)
59 \( 1 + (-46.5 + 26.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-8 + 13.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-100.5 + 58.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 25T + 5.32e3T^{2} \)
79 \( 1 + (-24 - 13.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (30 + 17.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 2T + 7.92e3T^{2} \)
97 \( 1 + (-21.5 + 37.2i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.85961218737605431131420894618, −12.17943238780265407766427289870, −11.40299044144793240646202092613, −10.11747961441074300386019301745, −9.550397450611721853186796576782, −8.238224340695612330926027781716, −6.82709696882981068985607989581, −6.22553180476418329971461814382, −3.71305706960788304913441990174, −1.80570243621156538987813081825, 1.06738850104828374950707415430, 3.20438836984961025217929012962, 5.54923994748377296817078999355, 6.59683339993252342435269938996, 8.191583940616612312323452113217, 8.934177262867916437164762452338, 9.961368820786361898919390258136, 11.01480917134874881744645439968, 12.20499939245017695472547043173, 13.09562163463974244992776810633

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