Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.612 - 0.790i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.93 + 2.06i)2-s + (−0.538 + 7.98i)4-s + (4.71 + 2.72i)5-s + (−20.9 + 12.0i)7-s + (−17.5 + 14.3i)8-s + (3.48 + 14.9i)10-s + (25.3 + 43.9i)11-s + (25.0 − 43.4i)13-s + (−65.4 − 19.9i)14-s + (−63.4 − 8.60i)16-s + 51.7i·17-s + 27.9i·19-s + (−24.2 + 36.1i)20-s + (−41.8 + 137. i)22-s + (−3.93 + 6.81i)23-s + ⋯
L(s)  = 1  + (0.682 + 0.730i)2-s + (−0.0673 + 0.997i)4-s + (0.421 + 0.243i)5-s + (−1.13 + 0.652i)7-s + (−0.774 + 0.632i)8-s + (0.110 + 0.474i)10-s + (0.696 + 1.20i)11-s + (0.535 − 0.927i)13-s + (−1.24 − 0.380i)14-s + (−0.990 − 0.134i)16-s + 0.737i·17-s + 0.337i·19-s + (−0.271 + 0.404i)20-s + (−0.405 + 1.33i)22-s + (−0.0356 + 0.0617i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.612 - 0.790i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (71, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.612 - 0.790i)\)
\(L(2)\)  \(\approx\)  \(0.890455 + 1.81547i\)
\(L(\frac12)\)  \(\approx\)  \(0.890455 + 1.81547i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-1.93 - 2.06i)T \)
3 \( 1 \)
good5 \( 1 + (-4.71 - 2.72i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (20.9 - 12.0i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-25.3 - 43.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-25.0 + 43.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 51.7iT - 4.91e3T^{2} \)
19 \( 1 - 27.9iT - 6.85e3T^{2} \)
23 \( 1 + (3.93 - 6.81i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-212. + 122. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-51.4 - 29.6i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 295.T + 5.06e4T^{2} \)
41 \( 1 + (-146. - 84.7i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-284. + 164. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (47.9 + 83.0i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 300. iT - 1.48e5T^{2} \)
59 \( 1 + (-113. + 196. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-173. - 300. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (904. + 522. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 243.T + 3.57e5T^{2} \)
73 \( 1 + 1.09e3T + 3.89e5T^{2} \)
79 \( 1 + (-530. + 306. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-283. - 490. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 212. iT - 7.04e5T^{2} \)
97 \( 1 + (-234. - 405. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.58179366904546836448200377534, −12.66006690032844905452119836216, −12.02156967750254687163352506065, −10.26305198958773583163119823537, −9.213403570269077616368874638893, −7.919515787129410496564765560683, −6.49052714580195235265132838775, −5.89966660855544089632787203422, −4.19048908106014153658427575594, −2.70361994164753761677799388489, 0.958069253197356653259170408975, 3.04414512828299490666287421684, 4.25679272875190858446038039496, 5.91408430564476792560438587659, 6.77270057972626462021200361771, 8.970461410337169159811043019100, 9.709138527827778754303267391234, 10.92577190041303117391683844547, 11.79623893019305300802965495038, 13.06775843872980427649189783238

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