Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.905 + 0.423i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.51 + 2.38i)2-s + (−3.38 − 7.24i)4-s + 13.1i·5-s + 4.49i·7-s + (22.4 + 2.92i)8-s + (−31.4 − 20.0i)10-s − 22.3·11-s − 73.5·13-s + (−10.7 − 6.81i)14-s + (−41.0 + 49.1i)16-s + 42.6i·17-s − 122. i·19-s + (95.6 − 44.7i)20-s + (33.9 − 53.3i)22-s − 197.·23-s + ⋯
L(s)  = 1  + (−0.536 + 0.843i)2-s + (−0.423 − 0.905i)4-s + 1.18i·5-s + 0.242i·7-s + (0.991 + 0.129i)8-s + (−0.995 − 0.633i)10-s − 0.612·11-s − 1.56·13-s + (−0.204 − 0.130i)14-s + (−0.641 + 0.767i)16-s + 0.608i·17-s − 1.47i·19-s + (1.06 − 0.499i)20-s + (0.328 − 0.516i)22-s − 1.79·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.423i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.905 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.905 + 0.423i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.905 + 0.423i)\)
\(L(2)\)  \(\approx\)  \(0.100807 - 0.453699i\)
\(L(\frac12)\)  \(\approx\)  \(0.100807 - 0.453699i\)
\(L(\frac{5}{2})\)   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 \( 1 + (1.51 - 2.38i)T \)
3 \( 1 \)
good5 \( 1 - 13.1iT - 125T^{2} \)
7 \( 1 - 4.49iT - 343T^{2} \)
11 \( 1 + 22.3T + 1.33e3T^{2} \)
13 \( 1 + 73.5T + 2.19e3T^{2} \)
17 \( 1 - 42.6iT - 4.91e3T^{2} \)
19 \( 1 + 122. iT - 6.85e3T^{2} \)
23 \( 1 + 197.T + 1.21e4T^{2} \)
29 \( 1 + 14.6iT - 2.43e4T^{2} \)
31 \( 1 - 147. iT - 2.97e4T^{2} \)
37 \( 1 - 234.T + 5.06e4T^{2} \)
41 \( 1 - 396. iT - 6.89e4T^{2} \)
43 \( 1 - 280. iT - 7.95e4T^{2} \)
47 \( 1 + 534.T + 1.03e5T^{2} \)
53 \( 1 - 337. iT - 1.48e5T^{2} \)
59 \( 1 - 672.T + 2.05e5T^{2} \)
61 \( 1 + 80.8T + 2.26e5T^{2} \)
67 \( 1 + 251. iT - 3.00e5T^{2} \)
71 \( 1 - 95.8T + 3.57e5T^{2} \)
73 \( 1 + 251.T + 3.89e5T^{2} \)
79 \( 1 - 499. iT - 4.93e5T^{2} \)
83 \( 1 + 16.1T + 5.71e5T^{2} \)
89 \( 1 + 321. iT - 7.04e5T^{2} \)
97 \( 1 + 210.T + 9.12e5T^{2} \)
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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

−14.24287561661442316167694017994, −12.99161268455469738755032259379, −11.46346673348884005223846145286, −10.35764361868191430653974694796, −9.623700699131690893608242136043, −8.128642560735445890513370894493, −7.18923600271869084016677010456, −6.19330570728805026350496955696, −4.76481364762127025197203485397, −2.53346962644923761559213104593, 0.28910281676697173826437296512, 2.13797049366818692189561971823, 4.07821942464880267484139673298, 5.29795142685982665320218437640, 7.52040831500939099157938011744, 8.339019613887686014935143447862, 9.625660040261528309688393723729, 10.23233163521911411090228500550, 11.82274394518426878739969787705, 12.37790298742563417471202283980

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