Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.566 - 0.823i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.99 + 0.195i)3-s + (−2.50 + 6.87i)5-s + (−1.62 + 9.22i)7-s + (8.92 + 1.16i)9-s + (−4.02 − 11.0i)11-s + (17.4 − 14.6i)13-s + (−8.82 + 20.0i)15-s + (−13.5 + 7.81i)17-s + (9.08 − 15.7i)19-s + (−6.67 + 27.3i)21-s + (23.0 − 4.06i)23-s + (−21.8 − 18.3i)25-s + (26.4 + 5.24i)27-s + (−24.8 + 29.6i)29-s + (−4.47 − 25.3i)31-s + ⋯
L(s)  = 1  + (0.997 + 0.0651i)3-s + (−0.500 + 1.37i)5-s + (−0.232 + 1.31i)7-s + (0.991 + 0.129i)9-s + (−0.366 − 1.00i)11-s + (1.34 − 1.12i)13-s + (−0.588 + 1.33i)15-s + (−0.795 + 0.459i)17-s + (0.478 − 0.828i)19-s + (−0.317 + 1.30i)21-s + (1.00 − 0.176i)23-s + (−0.872 − 0.732i)25-s + (0.980 + 0.194i)27-s + (−0.858 + 1.02i)29-s + (−0.144 − 0.818i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.823i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.566 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.566 - 0.823i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (65, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.566 - 0.823i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.49359 + 0.785270i\)
\(L(\frac12)\)  \(\approx\)  \(1.49359 + 0.785270i\)
\(L(2)\)   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 \( 1 + (-2.99 - 0.195i)T \)
good5 \( 1 + (2.50 - 6.87i)T + (-19.1 - 16.0i)T^{2} \)
7 \( 1 + (1.62 - 9.22i)T + (-46.0 - 16.7i)T^{2} \)
11 \( 1 + (4.02 + 11.0i)T + (-92.6 + 77.7i)T^{2} \)
13 \( 1 + (-17.4 + 14.6i)T + (29.3 - 166. i)T^{2} \)
17 \( 1 + (13.5 - 7.81i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-9.08 + 15.7i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-23.0 + 4.06i)T + (497. - 180. i)T^{2} \)
29 \( 1 + (24.8 - 29.6i)T + (-146. - 828. i)T^{2} \)
31 \( 1 + (4.47 + 25.3i)T + (-903. + 328. i)T^{2} \)
37 \( 1 + (-1.45 - 2.52i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (26.1 + 31.1i)T + (-291. + 1.65e3i)T^{2} \)
43 \( 1 + (-35.7 + 12.9i)T + (1.41e3 - 1.18e3i)T^{2} \)
47 \( 1 + (18.5 + 3.26i)T + (2.07e3 + 755. i)T^{2} \)
53 \( 1 + 12.3iT - 2.80e3T^{2} \)
59 \( 1 + (-4.38 + 12.0i)T + (-2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (15.6 - 88.8i)T + (-3.49e3 - 1.27e3i)T^{2} \)
67 \( 1 + (6.68 - 5.61i)T + (779. - 4.42e3i)T^{2} \)
71 \( 1 + (-81.5 + 47.0i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (25.3 - 43.8i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (78.4 + 65.8i)T + (1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (12.1 - 14.4i)T + (-1.19e3 - 6.78e3i)T^{2} \)
89 \( 1 + (52.8 + 30.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (139. - 50.8i)T + (7.20e3 - 6.04e3i)T^{2} \)
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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

−13.62667910595696947503250708536, −12.85750780108722114337224156458, −11.24411158497135120144914057161, −10.64247782294431662625381631497, −9.046708086356018405190891820798, −8.328032473516741600721052778326, −7.04398796618960224894304939134, −5.74390645702711255276262332204, −3.47072390490852602315317477392, −2.72506295615532799070816481250, 1.39038270311144851517775324006, 3.82854585032189135830094585116, 4.62556649620947237602459034491, 6.90475370910270476771353572397, 7.889875216426754962239881569544, 8.944249951509961787623442039380, 9.792772992871251968071368520320, 11.23363905708291860222406048631, 12.61992704642016861152654507727, 13.32748681787424041084032739451

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