Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.316 - 0.948i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.73 + 2.16i)3-s + (9.54 + 26.2i)5-s + (−0.541 − 3.07i)7-s + (71.6 + 37.8i)9-s + (−44.3 + 121. i)11-s + (−30.3 − 25.5i)13-s + (26.5 + 249. i)15-s + (−72.0 − 41.5i)17-s + (32.6 + 56.5i)19-s + (1.92 − 28.0i)21-s + (346. + 61.0i)23-s + (−117. + 98.5i)25-s + (543. + 485. i)27-s + (421. + 501. i)29-s + (−18.7 + 106. i)31-s + ⋯
L(s)  = 1  + (0.970 + 0.240i)3-s + (0.381 + 1.04i)5-s + (−0.0110 − 0.0627i)7-s + (0.884 + 0.467i)9-s + (−0.366 + 1.00i)11-s + (−0.179 − 0.150i)13-s + (0.118 + 1.10i)15-s + (−0.249 − 0.143i)17-s + (0.0904 + 0.156i)19-s + (0.00436 − 0.0635i)21-s + (0.654 + 0.115i)23-s + (−0.187 + 0.157i)25-s + (0.745 + 0.666i)27-s + (0.500 + 0.596i)29-s + (−0.0194 + 0.110i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.316 - 0.948i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.316 - 0.948i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.99804 + 1.43985i\)
\(L(\frac12)\)  \(\approx\)  \(1.99804 + 1.43985i\)
\(L(3)\)   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 \( 1 + (-8.73 - 2.16i)T \)
good5 \( 1 + (-9.54 - 26.2i)T + (-478. + 401. i)T^{2} \)
7 \( 1 + (0.541 + 3.07i)T + (-2.25e3 + 821. i)T^{2} \)
11 \( 1 + (44.3 - 121. i)T + (-1.12e4 - 9.41e3i)T^{2} \)
13 \( 1 + (30.3 + 25.5i)T + (4.95e3 + 2.81e4i)T^{2} \)
17 \( 1 + (72.0 + 41.5i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-32.6 - 56.5i)T + (-6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-346. - 61.0i)T + (2.62e5 + 9.57e4i)T^{2} \)
29 \( 1 + (-421. - 501. i)T + (-1.22e5 + 6.96e5i)T^{2} \)
31 \( 1 + (18.7 - 106. i)T + (-8.67e5 - 3.15e5i)T^{2} \)
37 \( 1 + (745. - 1.29e3i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + (-1.34e3 + 1.59e3i)T + (-4.90e5 - 2.78e6i)T^{2} \)
43 \( 1 + (1.40e3 + 511. i)T + (2.61e6 + 2.19e6i)T^{2} \)
47 \( 1 + (-2.36e3 + 416. i)T + (4.58e6 - 1.66e6i)T^{2} \)
53 \( 1 + 2.34e3iT - 7.89e6T^{2} \)
59 \( 1 + (978. + 2.68e3i)T + (-9.28e6 + 7.78e6i)T^{2} \)
61 \( 1 + (1.13e3 + 6.46e3i)T + (-1.30e7 + 4.73e6i)T^{2} \)
67 \( 1 + (6.61e3 + 5.54e3i)T + (3.49e6 + 1.98e7i)T^{2} \)
71 \( 1 + (426. + 245. i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (2.00e3 + 3.47e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (5.37e3 - 4.50e3i)T + (6.76e6 - 3.83e7i)T^{2} \)
83 \( 1 + (-1.60e3 - 1.91e3i)T + (-8.24e6 + 4.67e7i)T^{2} \)
89 \( 1 + (-1.29e4 + 7.46e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (-1.58e4 - 5.77e3i)T + (6.78e7 + 5.69e7i)T^{2} \)
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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

−13.43306744511473562199000443541, −12.31394272609731345808689209340, −10.70007988692356422512732539056, −10.07087169719432881220770637781, −8.968314257035727580660223407518, −7.59392616774786879636885033594, −6.74010340063684153994560463904, −4.87609075809365941701091317990, −3.27995509445915042039511895455, −2.11490853659349884716606318102, 1.07027426412423050308933742338, 2.73994271927428490758040759082, 4.39332199220652339478642596808, 5.85624791024350711161473459975, 7.43433294315008576125272045437, 8.646052986449283124620362212193, 9.159679523539565737006897477494, 10.49533836095236063859289711709, 11.99408568974110036959862076906, 13.08521035995730531410212713064

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