Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.982 - 0.187i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.59 + 1.20i)2-s + (1.08 − 3.84i)4-s + (−1.10 + 1.90i)5-s + (−7.23 + 4.17i)7-s + (2.90 + 7.45i)8-s + (−0.544 − 4.36i)10-s + (−4.54 + 2.62i)11-s + (−7.37 + 12.7i)13-s + (6.50 − 15.3i)14-s + (−13.6 − 8.38i)16-s − 28.2·17-s − 19.1i·19-s + (6.13 + 6.31i)20-s + (4.08 − 9.67i)22-s + (3.16 + 1.82i)23-s + ⋯
L(s)  = 1  + (−0.797 + 0.603i)2-s + (0.272 − 0.962i)4-s + (−0.220 + 0.381i)5-s + (−1.03 + 0.597i)7-s + (0.363 + 0.931i)8-s + (−0.0544 − 0.436i)10-s + (−0.413 + 0.238i)11-s + (−0.567 + 0.982i)13-s + (0.464 − 1.09i)14-s + (−0.851 − 0.524i)16-s − 1.66·17-s − 1.00i·19-s + (0.306 + 0.315i)20-s + (0.185 − 0.439i)22-s + (0.137 + 0.0794i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.982 - 0.187i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (91, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.982 - 0.187i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0373875 + 0.394775i\)
\(L(\frac12)\)  \(\approx\)  \(0.0373875 + 0.394775i\)
\(L(2)\)   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 \( 1 + (1.59 - 1.20i)T \)
3 \( 1 \)
good5 \( 1 + (1.10 - 1.90i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (7.23 - 4.17i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (4.54 - 2.62i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (7.37 - 12.7i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 28.2T + 289T^{2} \)
19 \( 1 + 19.1iT - 361T^{2} \)
23 \( 1 + (-3.16 - 1.82i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-12.3 - 21.3i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-32.9 - 19.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 4.21T + 1.36e3T^{2} \)
41 \( 1 + (-9.92 + 17.1i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (20.1 - 11.6i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-25.8 + 14.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 32.1T + 2.80e3T^{2} \)
59 \( 1 + (7.96 + 4.59i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (40.8 + 70.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (6.86 + 3.96i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 62.9iT - 5.04e3T^{2} \)
73 \( 1 - 33.3T + 5.32e3T^{2} \)
79 \( 1 + (53.7 - 31.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (103. - 59.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 107.T + 7.92e3T^{2} \)
97 \( 1 + (-1.78 - 3.09i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−14.14397431452105032447799757984, −12.99620263899121984917900212318, −11.63608874764082850981875175641, −10.61127215329244672586296935922, −9.420179598908872560204908855687, −8.739234920879475064913645635378, −7.08813097313343117666147902313, −6.52183259950896629279531679669, −4.87896815938109636284353184176, −2.54290796454051181040245522256, 0.35416633113750459101995979611, 2.77561568027857382393465918018, 4.28275385481656016629479970354, 6.39911068071294368710198447738, 7.68328696435862355959473585585, 8.695243825894644731015871414434, 9.948990250753907599779488107613, 10.57704879057864573792460871540, 11.90753415405360607542291346670, 12.86424490970715058158173494695

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