Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.976 - 0.214i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.81 − 0.285i)2-s + (7.83 − 1.60i)4-s + (14.4 + 8.36i)5-s + (−16.7 + 9.65i)7-s + (21.5 − 6.74i)8-s + (43.1 + 19.4i)10-s + (−2.44 − 4.22i)11-s + (6.03 − 10.4i)13-s + (−44.2 + 31.9i)14-s + (58.8 − 25.1i)16-s − 71.2i·17-s + 68.3i·19-s + (127. + 42.3i)20-s + (−8.07 − 11.2i)22-s + (68.0 − 117. i)23-s + ⋯
L(s)  = 1  + (0.994 − 0.100i)2-s + (0.979 − 0.200i)4-s + (1.29 + 0.748i)5-s + (−0.902 + 0.521i)7-s + (0.954 − 0.298i)8-s + (1.36 + 0.613i)10-s + (−0.0669 − 0.115i)11-s + (0.128 − 0.223i)13-s + (−0.845 + 0.609i)14-s + (0.919 − 0.392i)16-s − 1.01i·17-s + 0.824i·19-s + (1.41 + 0.473i)20-s + (−0.0782 − 0.108i)22-s + (0.616 − 1.06i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.976 - 0.214i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (71, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.976 - 0.214i)\)
\(L(2)\)  \(\approx\)  \(3.16202 + 0.343496i\)
\(L(\frac12)\)  \(\approx\)  \(3.16202 + 0.343496i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-2.81 + 0.285i)T \)
3 \( 1 \)
good5 \( 1 + (-14.4 - 8.36i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (16.7 - 9.65i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (2.44 + 4.22i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-6.03 + 10.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 71.2iT - 4.91e3T^{2} \)
19 \( 1 - 68.3iT - 6.85e3T^{2} \)
23 \( 1 + (-68.0 + 117. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (190. - 109. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (285. + 164. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 133.T + 5.06e4T^{2} \)
41 \( 1 + (-29.5 - 17.0i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (0.558 - 0.322i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (93.4 + 161. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 266. iT - 1.48e5T^{2} \)
59 \( 1 + (-104. + 180. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-0.801 - 1.38i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-371. - 214. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 386.T + 3.57e5T^{2} \)
73 \( 1 + 776.T + 3.89e5T^{2} \)
79 \( 1 + (-68.5 + 39.5i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-462. - 801. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.04e3iT - 7.04e5T^{2} \)
97 \( 1 + (-733. - 1.26e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.23371712212639422546308335360, −12.57384711931299123875446213663, −11.18727128590288978851651824927, −10.22733020178283534996481509809, −9.266679369397060511311588266005, −7.19569356433683287740339919101, −6.18496948205364087401136376363, −5.38417863194738593221030301230, −3.34815641854361311941406008113, −2.21093546168783796977701253188, 1.78071700580366098120759164201, 3.59439699503097070219424423668, 5.14954471095484840881754337212, 6.11359779391283238484857972975, 7.23061134994590304564055111941, 9.012921025679973797780724156810, 10.05001718693125450874257381712, 11.18471684126352424868795391490, 12.75101999164878670097700521026, 13.10592010905352237679836256259

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