Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.616 + 0.787i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.65 + 4.31i)2-s + (−5.21 − 31.5i)4-s + (51.8 − 29.9i)5-s + (33.8 + 19.5i)7-s + (155. + 93.0i)8-s + (−60.6 + 333. i)10-s + (−108. + 187. i)11-s + (−532. − 921. i)13-s + (−208. + 74.5i)14-s + (−969. + 329. i)16-s − 315. i·17-s − 1.88e3i·19-s + (−1.21e3 − 1.48e3i)20-s + (−413. − 1.15e3i)22-s + (1.20e3 + 2.07e3i)23-s + ⋯
L(s)  = 1  + (−0.646 + 0.762i)2-s + (−0.162 − 0.986i)4-s + (0.927 − 0.535i)5-s + (0.261 + 0.150i)7-s + (0.857 + 0.514i)8-s + (−0.191 + 1.05i)10-s + (−0.270 + 0.468i)11-s + (−0.873 − 1.51i)13-s + (−0.284 + 0.101i)14-s + (−0.946 + 0.321i)16-s − 0.264i·17-s − 1.20i·19-s + (−0.679 − 0.828i)20-s + (−0.182 − 0.509i)22-s + (0.473 + 0.819i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.616 + 0.787i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (35, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.616 + 0.787i)\)
\(L(3)\)  \(\approx\)  \(1.09824 - 0.535225i\)
\(L(\frac12)\)  \(\approx\)  \(1.09824 - 0.535225i\)
\(L(\frac{7}{2})\)   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 \( 1 + (3.65 - 4.31i)T \)
3 \( 1 \)
good5 \( 1 + (-51.8 + 29.9i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-33.8 - 19.5i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (108. - 187. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (532. + 921. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 315. iT - 1.41e6T^{2} \)
19 \( 1 + 1.88e3iT - 2.47e6T^{2} \)
23 \( 1 + (-1.20e3 - 2.07e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (5.18e3 + 2.99e3i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-6.94e3 + 4.01e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 5.24e3T + 6.93e7T^{2} \)
41 \( 1 + (-7.58e3 + 4.38e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (8.78e3 + 5.06e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-7.27e3 + 1.25e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 1.96e4iT - 4.18e8T^{2} \)
59 \( 1 + (-909. - 1.57e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.44e4 + 4.22e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (3.70e4 - 2.14e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 2.94e4T + 1.80e9T^{2} \)
73 \( 1 - 3.80e4T + 2.07e9T^{2} \)
79 \( 1 + (-1.28e4 - 7.42e3i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (6.01e4 - 1.04e5i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 3.54e4iT - 5.58e9T^{2} \)
97 \( 1 + (-1.49e3 + 2.58e3i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−12.95522705905264025928892042035, −11.33952398088167511281157892993, −10.00603300253076717838448554669, −9.452768892476991659966254891581, −8.184457770499941377096694414717, −7.16190504069130588089212805311, −5.64749459172410357374431136362, −4.97172469626514675873787636096, −2.25296664183579257935859603346, −0.58063083255162546526147907159, 1.55260408130598542629138314228, 2.75631205144292231912422622949, 4.44634809500999368540870846045, 6.28281272840010627851446776310, 7.53510177150011223687640124137, 8.852752769474680355651002032612, 9.848675951976744446065539952792, 10.64795399639654695928115204911, 11.68565747193593743596503339798, 12.74247290465552753515403678843

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