Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.122 + 0.992i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.02 − 2.82i)3-s + (1.26 − 3.47i)5-s + (−0.0728 + 0.412i)7-s + (−6.91 − 5.75i)9-s + (−1.69 − 4.64i)11-s + (3.65 − 3.06i)13-s + (−8.52 − 7.12i)15-s + (20.4 − 11.8i)17-s + (−13.5 + 23.4i)19-s + (1.09 + 0.626i)21-s + (20.7 − 3.66i)23-s + (8.64 + 7.25i)25-s + (−23.3 + 13.6i)27-s + (−2.12 + 2.53i)29-s + (3.01 + 17.0i)31-s + ⋯
L(s)  = 1  + (0.340 − 0.940i)3-s + (0.253 − 0.695i)5-s + (−0.0104 + 0.0589i)7-s + (−0.768 − 0.639i)9-s + (−0.153 − 0.422i)11-s + (0.281 − 0.236i)13-s + (−0.568 − 0.474i)15-s + (1.20 − 0.696i)17-s + (−0.712 + 1.23i)19-s + (0.0519 + 0.0298i)21-s + (0.902 − 0.159i)23-s + (0.345 + 0.290i)25-s + (−0.862 + 0.505i)27-s + (−0.0732 + 0.0873i)29-s + (0.0972 + 0.551i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.122 + 0.992i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (65, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.122 + 0.992i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.12000 - 0.990674i\)
\(L(\frac12)\)  \(\approx\)  \(1.12000 - 0.990674i\)
\(L(2)\)   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 \( 1 + (-1.02 + 2.82i)T \)
good5 \( 1 + (-1.26 + 3.47i)T + (-19.1 - 16.0i)T^{2} \)
7 \( 1 + (0.0728 - 0.412i)T + (-46.0 - 16.7i)T^{2} \)
11 \( 1 + (1.69 + 4.64i)T + (-92.6 + 77.7i)T^{2} \)
13 \( 1 + (-3.65 + 3.06i)T + (29.3 - 166. i)T^{2} \)
17 \( 1 + (-20.4 + 11.8i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (13.5 - 23.4i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-20.7 + 3.66i)T + (497. - 180. i)T^{2} \)
29 \( 1 + (2.12 - 2.53i)T + (-146. - 828. i)T^{2} \)
31 \( 1 + (-3.01 - 17.0i)T + (-903. + 328. i)T^{2} \)
37 \( 1 + (-24.9 - 43.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-26.0 - 31.0i)T + (-291. + 1.65e3i)T^{2} \)
43 \( 1 + (61.2 - 22.2i)T + (1.41e3 - 1.18e3i)T^{2} \)
47 \( 1 + (26.9 + 4.75i)T + (2.07e3 + 755. i)T^{2} \)
53 \( 1 + 59.1iT - 2.80e3T^{2} \)
59 \( 1 + (16.8 - 46.1i)T + (-2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (-11.9 + 67.9i)T + (-3.49e3 - 1.27e3i)T^{2} \)
67 \( 1 + (-56.4 + 47.3i)T + (779. - 4.42e3i)T^{2} \)
71 \( 1 + (-88.6 + 51.1i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (3.81 - 6.60i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (99.5 + 83.5i)T + (1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (44.1 - 52.5i)T + (-1.19e3 - 6.78e3i)T^{2} \)
89 \( 1 + (137. + 79.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (8.07 - 2.93i)T + (7.20e3 - 6.04e3i)T^{2} \)
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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

−13.09510153055297634811285239116, −12.45792738270493641840339465325, −11.35806608891302891587312545080, −9.861203415864414718082833423771, −8.653888959858685636015378786871, −7.85845334103328468994484984185, −6.44293806087343719212466543233, −5.24373972275880518617995951328, −3.16701476798465866382943990143, −1.23948316889773200032853019057, 2.65915800927611894767760996875, 4.10264690347516952543793686328, 5.55405950309240637577440432974, 7.02425957050938910768951444992, 8.433080611622334874478901322279, 9.555605518413973600952336801590, 10.50182775821491672422549622435, 11.25930520252607700142742624006, 12.77653073561673812992059351256, 13.94262290535049067292450350511

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