Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.531 + 0.847i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.48 − 1.40i)2-s + (28.0 + 15.3i)4-s + (77.8 − 44.9i)5-s + (−124. − 71.6i)7-s + (−132. − 123. i)8-s + (−489. + 137. i)10-s + (278. − 481. i)11-s + (179. + 311. i)13-s + (579. + 566. i)14-s + (551. + 863. i)16-s + 1.07e3i·17-s − 348. i·19-s + (2.87e3 − 64.1i)20-s + (−2.20e3 + 2.25e3i)22-s + (−1.46e3 − 2.54e3i)23-s + ⋯
L(s)  = 1  + (−0.968 − 0.248i)2-s + (0.876 + 0.480i)4-s + (1.39 − 0.803i)5-s + (−0.956 − 0.552i)7-s + (−0.730 − 0.683i)8-s + (−1.54 + 0.433i)10-s + (0.693 − 1.20i)11-s + (0.295 + 0.511i)13-s + (0.789 + 0.772i)14-s + (0.538 + 0.842i)16-s + 0.903i·17-s − 0.221i·19-s + (1.60 − 0.0358i)20-s + (−0.969 + 0.991i)22-s + (−0.579 − 1.00i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.531 + 0.847i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.531 + 0.847i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (35, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -0.531 + 0.847i)\)
\(L(3)\)  \(\approx\)  \(0.561719 - 1.01568i\)
\(L(\frac12)\)  \(\approx\)  \(0.561719 - 1.01568i\)
\(L(\frac{7}{2})\)   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 \( 1 + (5.48 + 1.40i)T \)
3 \( 1 \)
good5 \( 1 + (-77.8 + 44.9i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (124. + 71.6i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-278. + 481. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-179. - 311. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 1.07e3iT - 1.41e6T^{2} \)
19 \( 1 + 348. iT - 2.47e6T^{2} \)
23 \( 1 + (1.46e3 + 2.54e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (3.33e3 + 1.92e3i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (2.34e3 - 1.35e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 1.99e3T + 6.93e7T^{2} \)
41 \( 1 + (-1.34e4 + 7.75e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (1.24e4 + 7.20e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-1.23e3 + 2.13e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 1.33e4iT - 4.18e8T^{2} \)
59 \( 1 + (1.98e4 + 3.44e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (211. - 366. i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.40e4 + 8.09e3i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.90e3T + 1.80e9T^{2} \)
73 \( 1 + 1.10e4T + 2.07e9T^{2} \)
79 \( 1 + (-6.43e4 - 3.71e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (4.85e4 - 8.41e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 7.44e4iT - 5.58e9T^{2} \)
97 \( 1 + (-4.79e4 + 8.30e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−12.46302576474827104171848491414, −11.04738427139433754179079263528, −10.04702845329270974852019405772, −9.221631084689165572379481469223, −8.452170936620675874785928297790, −6.64131960353355876993205060363, −5.92602806389259442509766780477, −3.68121692405339250097202387973, −1.90745470664867267850874898977, −0.58754375194353222097981950573, 1.71109807213527851676387532027, 2.91925621112588827101907829953, 5.64430869697842445134739973217, 6.45807330485886701458470840505, 7.44590832871769979648959998464, 9.377870869531242839436788293619, 9.553718232763337723345281868538, 10.60188057665145714262751786097, 11.90025268689152044849147362319, 13.14229898907884952245902741328

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