Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.891 - 0.453i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0634 + 2.99i)3-s + (5.64 − 6.73i)5-s + (4.05 + 1.47i)7-s + (−8.99 + 0.380i)9-s + (12.6 + 15.0i)11-s + (−1.31 + 7.45i)13-s + (20.5 + 16.5i)15-s + (−3.16 − 1.82i)17-s + (−16.8 − 29.1i)19-s + (−4.16 + 12.2i)21-s + (1.05 + 2.88i)23-s + (−9.07 − 51.4i)25-s + (−1.71 − 26.9i)27-s + (23.9 − 4.22i)29-s + (−27.1 + 9.87i)31-s + ⋯
L(s)  = 1  + (0.0211 + 0.999i)3-s + (1.12 − 1.34i)5-s + (0.578 + 0.210i)7-s + (−0.999 + 0.0422i)9-s + (1.15 + 1.37i)11-s + (−0.101 + 0.573i)13-s + (1.37 + 1.10i)15-s + (−0.186 − 0.107i)17-s + (−0.885 − 1.53i)19-s + (−0.198 + 0.583i)21-s + (0.0457 + 0.125i)23-s + (−0.362 − 2.05i)25-s + (−0.0634 − 0.997i)27-s + (0.825 − 0.145i)29-s + (−0.875 + 0.318i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.891 - 0.453i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (41, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.891 - 0.453i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.59417 + 0.382276i\)
\(L(\frac12)\)  \(\approx\)  \(1.59417 + 0.382276i\)
\(L(2)\)   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 \( 1 \)
3 \( 1 + (-0.0634 - 2.99i)T \)
good5 \( 1 + (-5.64 + 6.73i)T + (-4.34 - 24.6i)T^{2} \)
7 \( 1 + (-4.05 - 1.47i)T + (37.5 + 31.4i)T^{2} \)
11 \( 1 + (-12.6 - 15.0i)T + (-21.0 + 119. i)T^{2} \)
13 \( 1 + (1.31 - 7.45i)T + (-158. - 57.8i)T^{2} \)
17 \( 1 + (3.16 + 1.82i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (16.8 + 29.1i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-1.05 - 2.88i)T + (-405. + 340. i)T^{2} \)
29 \( 1 + (-23.9 + 4.22i)T + (790. - 287. i)T^{2} \)
31 \( 1 + (27.1 - 9.87i)T + (736. - 617. i)T^{2} \)
37 \( 1 + (14.9 - 25.8i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (22.6 + 3.99i)T + (1.57e3 + 574. i)T^{2} \)
43 \( 1 + (6.85 - 5.74i)T + (321. - 1.82e3i)T^{2} \)
47 \( 1 + (-3.60 + 9.90i)T + (-1.69e3 - 1.41e3i)T^{2} \)
53 \( 1 + 70.8iT - 2.80e3T^{2} \)
59 \( 1 + (43.8 - 52.3i)T + (-604. - 3.42e3i)T^{2} \)
61 \( 1 + (99.7 + 36.3i)T + (2.85e3 + 2.39e3i)T^{2} \)
67 \( 1 + (4.27 - 24.2i)T + (-4.21e3 - 1.53e3i)T^{2} \)
71 \( 1 + (-29.9 - 17.2i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-20.9 - 36.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-8.78 - 49.8i)T + (-5.86e3 + 2.13e3i)T^{2} \)
83 \( 1 + (30.9 - 5.45i)T + (6.47e3 - 2.35e3i)T^{2} \)
89 \( 1 + (-40.0 + 23.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-84.8 + 71.1i)T + (1.63e3 - 9.26e3i)T^{2} \)
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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

−13.63632916371203998139892559573, −12.46496647383466965458684736955, −11.47920647706121123756753757238, −10.05473976808442885931030238549, −9.207766860152375932459928255334, −8.673534138565565224963496370400, −6.57404402486259309864469013932, −5.02298494918679044600906318915, −4.48215712246340466050579761150, −1.91753381216973833803724942828, 1.72299145083349154812555333483, 3.27785781283392090503580166437, 5.89750733646906174006112915476, 6.41913383154097591072946651723, 7.73224954052582079988333937558, 8.942468542052027223098869449359, 10.49061072906477773266076600557, 11.16016596912625526035467299693, 12.40483478816810983463993099752, 13.70989025775689634962175153275

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