Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.125 - 0.992i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.71 − 0.799i)2-s + (6.72 + 4.33i)4-s + (14.2 + 8.25i)5-s + (−19.2 + 11.1i)7-s + (−14.7 − 17.1i)8-s + (−32.1 − 33.8i)10-s + (−6.37 − 11.0i)11-s + (−11.1 + 19.3i)13-s + (61.1 − 14.7i)14-s + (26.3 + 58.3i)16-s + 117. i·17-s + 27.7i·19-s + (60.2 + 117. i)20-s + (8.46 + 35.0i)22-s + (−17.5 + 30.4i)23-s + ⋯
L(s)  = 1  + (−0.959 − 0.282i)2-s + (0.840 + 0.542i)4-s + (1.27 + 0.737i)5-s + (−1.04 + 0.600i)7-s + (−0.652 − 0.757i)8-s + (−1.01 − 1.06i)10-s + (−0.174 − 0.302i)11-s + (−0.238 + 0.413i)13-s + (1.16 − 0.281i)14-s + (0.411 + 0.911i)16-s + 1.67i·17-s + 0.334i·19-s + (0.673 + 1.31i)20-s + (0.0820 + 0.339i)22-s + (−0.159 + 0.276i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.125 - 0.992i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (71, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.125 - 0.992i)\)
\(L(2)\)  \(\approx\)  \(0.715658 + 0.630755i\)
\(L(\frac12)\)  \(\approx\)  \(0.715658 + 0.630755i\)
\(L(\frac{5}{2})\)   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 \( 1 + (2.71 + 0.799i)T \)
3 \( 1 \)
good5 \( 1 + (-14.2 - 8.25i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (19.2 - 11.1i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (6.37 + 11.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (11.1 - 19.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 117. iT - 4.91e3T^{2} \)
19 \( 1 - 27.7iT - 6.85e3T^{2} \)
23 \( 1 + (17.5 - 30.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (1.01 - 0.584i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-119. - 68.9i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 233.T + 5.06e4T^{2} \)
41 \( 1 + (13.2 + 7.65i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (361. - 208. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-116. - 201. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 180. iT - 1.48e5T^{2} \)
59 \( 1 + (-313. + 543. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (382. + 661. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-113. - 65.5i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 22.6T + 3.57e5T^{2} \)
73 \( 1 - 387.T + 3.89e5T^{2} \)
79 \( 1 + (-486. + 280. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-342. - 592. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 278. iT - 7.04e5T^{2} \)
97 \( 1 + (264. + 458. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.24801845531083810232398023211, −12.39565033111474089629162892666, −11.02591957357022081592319876932, −10.04107944677596512547083907744, −9.481924037263867398798375911660, −8.221119591629045731568158040515, −6.56461671450865670367216788917, −6.03011174918634606911280381517, −3.22489277596346533389681837767, −1.96769807473116773700288705308, 0.69317691044304699924153782926, 2.56171081373911547111668642267, 5.13262468378582484578749757049, 6.33348484895620806405382143473, 7.39994688889964671138714824328, 8.927705341309189908108829556108, 9.738965987684915611624780163621, 10.29403122024311009513322953345, 11.86460747561164508428432240796, 13.14448522188551747380927485227

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