Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.638 - 0.769i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.52 + 1.27i)2-s + (4.74 + 6.44i)4-s + (1.23 + 0.715i)5-s + (23.8 − 13.7i)7-s + (3.76 + 22.3i)8-s + (2.21 + 3.38i)10-s + (11.1 + 19.2i)11-s + (−34.5 + 59.9i)13-s + (77.8 − 4.33i)14-s + (−18.9 + 61.1i)16-s − 31.4i·17-s − 11.4i·19-s + (1.27 + 11.3i)20-s + (3.49 + 62.7i)22-s + (72.6 − 125. i)23-s + ⋯
L(s)  = 1  + (0.892 + 0.451i)2-s + (0.593 + 0.805i)4-s + (0.110 + 0.0639i)5-s + (1.28 − 0.744i)7-s + (0.166 + 0.986i)8-s + (0.0700 + 0.107i)10-s + (0.304 + 0.527i)11-s + (−0.738 + 1.27i)13-s + (1.48 − 0.0828i)14-s + (−0.296 + 0.955i)16-s − 0.448i·17-s − 0.138i·19-s + (0.0142 + 0.127i)20-s + (0.0338 + 0.608i)22-s + (0.658 − 1.14i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.638 - 0.769i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (71, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.638 - 0.769i)\)
\(L(2)\)  \(\approx\)  \(2.60854 + 1.22467i\)
\(L(\frac12)\)  \(\approx\)  \(2.60854 + 1.22467i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-2.52 - 1.27i)T \)
3 \( 1 \)
good5 \( 1 + (-1.23 - 0.715i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-23.8 + 13.7i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-11.1 - 19.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (34.5 - 59.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 31.4iT - 4.91e3T^{2} \)
19 \( 1 + 11.4iT - 6.85e3T^{2} \)
23 \( 1 + (-72.6 + 125. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-93.6 + 54.0i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (102. + 59.3i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 300.T + 5.06e4T^{2} \)
41 \( 1 + (344. + 199. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-173. + 100. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-151. - 262. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 243. iT - 1.48e5T^{2} \)
59 \( 1 + (-41.9 + 72.6i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-199. - 345. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (307. + 177. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 866.T + 3.57e5T^{2} \)
73 \( 1 - 64.6T + 3.89e5T^{2} \)
79 \( 1 + (-354. + 204. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (79.8 + 138. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.49e3iT - 7.04e5T^{2} \)
97 \( 1 + (-700. - 1.21e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.75743513165763028542874910927, −12.28983589307285525655668382710, −11.57515393479514742326277621708, −10.44716481694230994269743138907, −8.787156184611059171658155928063, −7.50168204590657573961726297990, −6.70360171715825005218570120734, −4.96447188116703260884229485925, −4.20950875223986617677649069203, −2.10342002309472312389137088771, 1.61137019726309867837660849835, 3.25120912389551870065749814006, 5.01000040294171697490456772419, 5.70070825214509117612122822751, 7.42847945965517475152770584310, 8.759252079663090049810770664206, 10.20513838299551435967766233650, 11.23931223956756275942963534271, 12.02425367882389251981515468532, 13.02408311634510141482059693218

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