Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.495 + 0.868i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.78 + 7.62i)3-s + (−3.22 − 8.85i)5-s + (−16.2 − 92.0i)7-s + (−35.1 + 72.9i)9-s + (74.5 − 204. i)11-s + (−77.8 − 65.2i)13-s + (52.0 − 66.9i)15-s + (135. + 78.4i)17-s + (237. + 412. i)19-s + (623. − 564. i)21-s + (50.4 + 8.89i)23-s + (410. − 344. i)25-s + (−724. + 81.2i)27-s + (−936. − 1.11e3i)29-s + (62.7 − 355. i)31-s + ⋯
L(s)  = 1  + (0.531 + 0.846i)3-s + (−0.128 − 0.354i)5-s + (−0.331 − 1.87i)7-s + (−0.434 + 0.900i)9-s + (0.616 − 1.69i)11-s + (−0.460 − 0.386i)13-s + (0.231 − 0.297i)15-s + (0.470 + 0.271i)17-s + (0.659 + 1.14i)19-s + (1.41 − 1.27i)21-s + (0.0953 + 0.0168i)23-s + (0.657 − 0.551i)25-s + (−0.993 + 0.111i)27-s + (−1.11 − 1.32i)29-s + (0.0653 − 0.370i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.495 + 0.868i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.495 + 0.868i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.49475 - 0.867604i\)
\(L(\frac12)\)  \(\approx\)  \(1.49475 - 0.867604i\)
\(L(3)\)   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 \( 1 + (-4.78 - 7.62i)T \)
good5 \( 1 + (3.22 + 8.85i)T + (-478. + 401. i)T^{2} \)
7 \( 1 + (16.2 + 92.0i)T + (-2.25e3 + 821. i)T^{2} \)
11 \( 1 + (-74.5 + 204. i)T + (-1.12e4 - 9.41e3i)T^{2} \)
13 \( 1 + (77.8 + 65.2i)T + (4.95e3 + 2.81e4i)T^{2} \)
17 \( 1 + (-135. - 78.4i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-237. - 412. i)T + (-6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-50.4 - 8.89i)T + (2.62e5 + 9.57e4i)T^{2} \)
29 \( 1 + (936. + 1.11e3i)T + (-1.22e5 + 6.96e5i)T^{2} \)
31 \( 1 + (-62.7 + 355. i)T + (-8.67e5 - 3.15e5i)T^{2} \)
37 \( 1 + (499. - 864. i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + (-1.27e3 + 1.51e3i)T + (-4.90e5 - 2.78e6i)T^{2} \)
43 \( 1 + (-739. - 269. i)T + (2.61e6 + 2.19e6i)T^{2} \)
47 \( 1 + (280. - 49.4i)T + (4.58e6 - 1.66e6i)T^{2} \)
53 \( 1 + 3.26e3iT - 7.89e6T^{2} \)
59 \( 1 + (-1.21e3 - 3.33e3i)T + (-9.28e6 + 7.78e6i)T^{2} \)
61 \( 1 + (-744. - 4.21e3i)T + (-1.30e7 + 4.73e6i)T^{2} \)
67 \( 1 + (-5.40e3 - 4.53e3i)T + (3.49e6 + 1.98e7i)T^{2} \)
71 \( 1 + (6.34e3 + 3.66e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (-851. - 1.47e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (4.66e3 - 3.91e3i)T + (6.76e6 - 3.83e7i)T^{2} \)
83 \( 1 + (-1.61e3 - 1.91e3i)T + (-8.24e6 + 4.67e7i)T^{2} \)
89 \( 1 + (-9.81e3 + 5.66e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (6.39e3 + 2.32e3i)T + (6.78e7 + 5.69e7i)T^{2} \)
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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

−13.14944253166498720963714472321, −11.51570877215083218037270426861, −10.49420270028016964192888861818, −9.768485045519320053418364460540, −8.436795549608905506456371674813, −7.50032918065626947970786167759, −5.79115257072139678024300744475, −4.14777566867802244592139517138, −3.39455730819902072096133670138, −0.74509906429027767125697930068, 1.91506631173588852328052737714, 3.02634334045743100516368597727, 5.16972674598565231370446931738, 6.65409172923865948797072506968, 7.46065248063599804663917134453, 9.096426943440341045669226685274, 9.406000534067491050537975442141, 11.45881104488337164257506453369, 12.32854904123299281771053708726, 12.82440678312811331228310238218

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