Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.525 + 0.850i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.72 − 0.772i)2-s + (6.80 − 4.20i)4-s − 3.33i·5-s − 16.9i·7-s + (15.2 − 16.6i)8-s + (−2.57 − 9.06i)10-s + 16.8·11-s + 25.0·13-s + (−13.0 − 45.9i)14-s + (28.6 − 57.2i)16-s + 116. i·17-s − 85.4i·19-s + (−14.0 − 22.6i)20-s + (45.7 − 13.0i)22-s − 158.·23-s + ⋯
L(s)  = 1  + (0.961 − 0.273i)2-s + (0.850 − 0.525i)4-s − 0.297i·5-s − 0.912i·7-s + (0.674 − 0.737i)8-s + (−0.0813 − 0.286i)10-s + 0.461·11-s + 0.535·13-s + (−0.249 − 0.877i)14-s + (0.447 − 0.894i)16-s + 1.66i·17-s − 1.03i·19-s + (−0.156 − 0.253i)20-s + (0.443 − 0.126i)22-s − 1.43·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.525 + 0.850i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.525 + 0.850i)\)
\(L(2)\)  \(\approx\)  \(2.47080 - 1.37798i\)
\(L(\frac12)\)  \(\approx\)  \(2.47080 - 1.37798i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-2.72 + 0.772i)T \)
3 \( 1 \)
good5 \( 1 + 3.33iT - 125T^{2} \)
7 \( 1 + 16.9iT - 343T^{2} \)
11 \( 1 - 16.8T + 1.33e3T^{2} \)
13 \( 1 - 25.0T + 2.19e3T^{2} \)
17 \( 1 - 116. iT - 4.91e3T^{2} \)
19 \( 1 + 85.4iT - 6.85e3T^{2} \)
23 \( 1 + 158.T + 1.21e4T^{2} \)
29 \( 1 - 269. iT - 2.43e4T^{2} \)
31 \( 1 - 36.0iT - 2.97e4T^{2} \)
37 \( 1 + 353.T + 5.06e4T^{2} \)
41 \( 1 - 144. iT - 6.89e4T^{2} \)
43 \( 1 - 368. iT - 7.95e4T^{2} \)
47 \( 1 - 397.T + 1.03e5T^{2} \)
53 \( 1 + 96.0iT - 1.48e5T^{2} \)
59 \( 1 + 294.T + 2.05e5T^{2} \)
61 \( 1 - 146.T + 2.26e5T^{2} \)
67 \( 1 - 301. iT - 3.00e5T^{2} \)
71 \( 1 + 687.T + 3.57e5T^{2} \)
73 \( 1 + 312.T + 3.89e5T^{2} \)
79 \( 1 + 602. iT - 4.93e5T^{2} \)
83 \( 1 - 1.33e3T + 5.71e5T^{2} \)
89 \( 1 + 856. iT - 7.04e5T^{2} \)
97 \( 1 - 218.T + 9.12e5T^{2} \)
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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

−13.06524435016397759371624937004, −12.23857400766702653799819696487, −10.96261980390955585204437661607, −10.30186908737400378021993857948, −8.691395733913569671172332835470, −7.16351928382697751957425006565, −6.09290115743264056093444280242, −4.59702100128028536896860709690, −3.52230809503140834792851672167, −1.42594036397854903215337505308, 2.34739037736927164646439130798, 3.84216745294029808907480061040, 5.39219324632049597344430279515, 6.37344836524403987143118955521, 7.64327030498012766015446843564, 8.932656937019682728818440945468, 10.42375665163589683799949611857, 11.82498717464430800665991741348, 12.13893374437878496790442400753, 13.66111602575227292163383566079

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