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  :  $\chi_{108} (5, \cdot )$
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

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

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