Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.900 - 0.434i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.66 − 4.46i)3-s + (−12.8 + 10.7i)5-s + (−20.8 + 7.57i)7-s + (−12.8 − 23.7i)9-s + (−41.5 − 34.8i)11-s + (12.2 + 69.4i)13-s + (13.8 + 85.7i)15-s + (17.7 + 30.7i)17-s + (37.9 − 65.7i)19-s + (−21.5 + 112. i)21-s + (−161. − 58.6i)23-s + (26.8 − 152. i)25-s + (−140. − 5.97i)27-s + (−25.1 + 142. i)29-s + (160. + 58.2i)31-s + ⋯
L(s)  = 1  + (0.512 − 0.858i)3-s + (−1.14 + 0.960i)5-s + (−1.12 + 0.408i)7-s + (−0.475 − 0.879i)9-s + (−1.13 − 0.955i)11-s + (0.261 + 1.48i)13-s + (0.238 + 1.47i)15-s + (0.252 + 0.437i)17-s + (0.458 − 0.793i)19-s + (−0.224 + 1.17i)21-s + (−1.46 − 0.531i)23-s + (0.214 − 1.21i)25-s + (−0.999 − 0.0425i)27-s + (−0.161 + 0.913i)29-s + (0.927 + 0.337i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.900 - 0.434i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (25, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.900 - 0.434i)\)
\(L(2)\)  \(\approx\)  \(0.0255686 + 0.111753i\)
\(L(\frac12)\)  \(\approx\)  \(0.0255686 + 0.111753i\)
\(L(\frac{5}{2})\)   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 \( 1 \)
3 \( 1 + (-2.66 + 4.46i)T \)
good5 \( 1 + (12.8 - 10.7i)T + (21.7 - 123. i)T^{2} \)
7 \( 1 + (20.8 - 7.57i)T + (262. - 220. i)T^{2} \)
11 \( 1 + (41.5 + 34.8i)T + (231. + 1.31e3i)T^{2} \)
13 \( 1 + (-12.2 - 69.4i)T + (-2.06e3 + 751. i)T^{2} \)
17 \( 1 + (-17.7 - 30.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-37.9 + 65.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (161. + 58.6i)T + (9.32e3 + 7.82e3i)T^{2} \)
29 \( 1 + (25.1 - 142. i)T + (-2.29e4 - 8.34e3i)T^{2} \)
31 \( 1 + (-160. - 58.2i)T + (2.28e4 + 1.91e4i)T^{2} \)
37 \( 1 + (180. + 313. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (24.4 + 138. i)T + (-6.47e4 + 2.35e4i)T^{2} \)
43 \( 1 + (-269. - 226. i)T + (1.38e4 + 7.82e4i)T^{2} \)
47 \( 1 + (74.2 - 27.0i)T + (7.95e4 - 6.67e4i)T^{2} \)
53 \( 1 + 195.T + 1.48e5T^{2} \)
59 \( 1 + (199. - 167. i)T + (3.56e4 - 2.02e5i)T^{2} \)
61 \( 1 + (-347. + 126. i)T + (1.73e5 - 1.45e5i)T^{2} \)
67 \( 1 + (13.6 + 77.4i)T + (-2.82e5 + 1.02e5i)T^{2} \)
71 \( 1 + (365. + 633. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (361. - 626. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-52.6 + 298. i)T + (-4.63e5 - 1.68e5i)T^{2} \)
83 \( 1 + (-8.20 + 46.5i)T + (-5.37e5 - 1.95e5i)T^{2} \)
89 \( 1 + (340. - 589. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-426. - 358. i)T + (1.58e5 + 8.98e5i)T^{2} \)
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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

−13.72845957597840060334242720185, −12.58846092198793833379198718352, −11.72621135449829442478232507751, −10.68435466084987824429580246968, −9.144721986668449298445540779323, −8.072153593349021481977289944117, −7.02963449174480722880592998556, −6.13429258695043977969656018504, −3.67747654164140928248709059929, −2.67367511052098168501207152144, 0.05605312044804790893152907321, 3.14062252044821812784274928929, 4.25825142682932179725610987498, 5.49875548386592945427615856719, 7.70385869284075100613761676002, 8.197644109473158596789012607564, 9.804549102612572258332063036479, 10.23875761547098544892563025996, 11.85963565969833979944453626498, 12.81504176615670175139239792342

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