Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.415 - 0.909i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.727 + 2.91i)3-s + (0.0686 − 0.188i)5-s + (−1.47 + 8.38i)7-s + (−7.94 − 4.23i)9-s + (2.58 + 7.09i)11-s + (−12.5 + 10.5i)13-s + (0.499 + 0.337i)15-s + (5.21 − 3.01i)17-s + (−0.189 + 0.328i)19-s + (−23.3 − 10.4i)21-s + (27.6 − 4.88i)23-s + (19.1 + 16.0i)25-s + (18.1 − 20.0i)27-s + (26.6 − 31.7i)29-s + (2.35 + 13.3i)31-s + ⋯
L(s)  = 1  + (−0.242 + 0.970i)3-s + (0.0137 − 0.0377i)5-s + (−0.211 + 1.19i)7-s + (−0.882 − 0.470i)9-s + (0.234 + 0.645i)11-s + (−0.966 + 0.810i)13-s + (0.0332 + 0.0224i)15-s + (0.306 − 0.177i)17-s + (−0.00999 + 0.0173i)19-s + (−1.11 − 0.495i)21-s + (1.20 − 0.212i)23-s + (0.764 + 0.641i)25-s + (0.670 − 0.741i)27-s + (0.918 − 1.09i)29-s + (0.0760 + 0.431i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.415 - 0.909i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (65, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.415 - 0.909i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.578341 + 0.900216i\)
\(L(\frac12)\)  \(\approx\)  \(0.578341 + 0.900216i\)
\(L(2)\)   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 \( 1 + (0.727 - 2.91i)T \)
good5 \( 1 + (-0.0686 + 0.188i)T + (-19.1 - 16.0i)T^{2} \)
7 \( 1 + (1.47 - 8.38i)T + (-46.0 - 16.7i)T^{2} \)
11 \( 1 + (-2.58 - 7.09i)T + (-92.6 + 77.7i)T^{2} \)
13 \( 1 + (12.5 - 10.5i)T + (29.3 - 166. i)T^{2} \)
17 \( 1 + (-5.21 + 3.01i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (0.189 - 0.328i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-27.6 + 4.88i)T + (497. - 180. i)T^{2} \)
29 \( 1 + (-26.6 + 31.7i)T + (-146. - 828. i)T^{2} \)
31 \( 1 + (-2.35 - 13.3i)T + (-903. + 328. i)T^{2} \)
37 \( 1 + (2.26 + 3.91i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (49.3 + 58.7i)T + (-291. + 1.65e3i)T^{2} \)
43 \( 1 + (-1.63 + 0.596i)T + (1.41e3 - 1.18e3i)T^{2} \)
47 \( 1 + (-75.3 - 13.2i)T + (2.07e3 + 755. i)T^{2} \)
53 \( 1 - 85.8iT - 2.80e3T^{2} \)
59 \( 1 + (-6.23 + 17.1i)T + (-2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (6.51 - 36.9i)T + (-3.49e3 - 1.27e3i)T^{2} \)
67 \( 1 + (53.9 - 45.2i)T + (779. - 4.42e3i)T^{2} \)
71 \( 1 + (38.9 - 22.4i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-51.2 + 88.7i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-64.7 - 54.2i)T + (1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (44.2 - 52.6i)T + (-1.19e3 - 6.78e3i)T^{2} \)
89 \( 1 + (119. + 68.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-112. + 41.0i)T + (7.20e3 - 6.04e3i)T^{2} \)
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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

−14.03913715748221677453029932736, −12.36106518117822070582195084169, −11.85564328223180417271692753128, −10.53438427018315189407321814764, −9.422898764137991610995702654814, −8.816482879812660826792511663217, −6.98208789870286827536641185316, −5.55991250337053252351310968894, −4.51432394016280081903954538064, −2.72242632410378928918706844726, 0.844588128641054475906723176594, 3.08992062303874412727627074590, 5.05969503859459061179486914790, 6.55986724533310171260287007411, 7.41204125469417893843095553692, 8.523783544941140652736712938033, 10.15837846571484109950163346495, 11.04322913515600496131178261736, 12.25734658470771812028168822469, 13.12559813909983629530087327147

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