Properties

Degree 2
Conductor $ 2^{4} \cdot 3 $
Sign $0.997 + 0.0701i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.78 + 0.911i)2-s + (1.22 − 1.22i)3-s + (2.33 − 3.24i)4-s + (1.00 − 1.00i)5-s + (−1.06 + 3.29i)6-s + 10.0·7-s + (−1.20 + 7.90i)8-s − 2.99i·9-s + (−0.875 + 2.71i)10-s + (2.26 + 2.26i)11-s + (−1.11 − 6.83i)12-s + (−6.88 − 6.88i)13-s + (−17.8 + 9.13i)14-s − 2.46i·15-s + (−5.07 − 15.1i)16-s − 22.3·17-s + ⋯
L(s)  = 1  + (−0.890 + 0.455i)2-s + (0.408 − 0.408i)3-s + (0.584 − 0.811i)4-s + (0.201 − 0.201i)5-s + (−0.177 + 0.549i)6-s + 1.43·7-s + (−0.150 + 0.988i)8-s − 0.333i·9-s + (−0.0875 + 0.271i)10-s + (0.205 + 0.205i)11-s + (−0.0926 − 0.569i)12-s + (−0.529 − 0.529i)13-s + (−1.27 + 0.652i)14-s − 0.164i·15-s + (−0.316 − 0.948i)16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0701i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(48\)    =    \(2^{4} \cdot 3\)
\( \varepsilon \)  =  $0.997 + 0.0701i$
motivic weight  =  \(2\)
character  :  $\chi_{48} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 48,\ (\ :1),\ 0.997 + 0.0701i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.949911 - 0.0333689i\)
\(L(\frac12)\)  \(\approx\)  \(0.949911 - 0.0333689i\)
\(L(2)\)   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 \( 1 + (1.78 - 0.911i)T \)
3 \( 1 + (-1.22 + 1.22i)T \)
good5 \( 1 + (-1.00 + 1.00i)T - 25iT^{2} \)
7 \( 1 - 10.0T + 49T^{2} \)
11 \( 1 + (-2.26 - 2.26i)T + 121iT^{2} \)
13 \( 1 + (6.88 + 6.88i)T + 169iT^{2} \)
17 \( 1 + 22.3T + 289T^{2} \)
19 \( 1 + (16.8 - 16.8i)T - 361iT^{2} \)
23 \( 1 - 33.2T + 529T^{2} \)
29 \( 1 + (24.6 + 24.6i)T + 841iT^{2} \)
31 \( 1 - 41.3iT - 961T^{2} \)
37 \( 1 + (6.60 - 6.60i)T - 1.36e3iT^{2} \)
41 \( 1 - 47.1iT - 1.68e3T^{2} \)
43 \( 1 + (48.8 + 48.8i)T + 1.84e3iT^{2} \)
47 \( 1 + 45.6iT - 2.20e3T^{2} \)
53 \( 1 + (-25.1 + 25.1i)T - 2.80e3iT^{2} \)
59 \( 1 + (-6.23 - 6.23i)T + 3.48e3iT^{2} \)
61 \( 1 + (-35.9 - 35.9i)T + 3.72e3iT^{2} \)
67 \( 1 + (-10.2 + 10.2i)T - 4.48e3iT^{2} \)
71 \( 1 - 11.9T + 5.04e3T^{2} \)
73 \( 1 + 111. iT - 5.32e3T^{2} \)
79 \( 1 + 4.46iT - 6.24e3T^{2} \)
83 \( 1 + (-10.1 + 10.1i)T - 6.88e3iT^{2} \)
89 \( 1 - 21.9iT - 7.92e3T^{2} \)
97 \( 1 - 107.T + 9.40e3T^{2} \)
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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

−15.13865659097150394382989108705, −14.73902201821293884395287115111, −13.29573342949317834423757883403, −11.68309274802240828880611444543, −10.55554166550474892854952761772, −9.012429471452659227751884744783, −8.120794609296671602233459707337, −6.91449198958805357920618074753, −5.11393796058501629962095839836, −1.83178116133491254186155661668, 2.24667172879860080816651333436, 4.49102081397836036664684980251, 6.99004006112537886485174397334, 8.413467398195352673657293578000, 9.276399313415607694901956751152, 10.83325635337146326300549005203, 11.39922327493327407769392687577, 13.07185284558743146232146104911, 14.50985868534496028549840764747, 15.42825390579123626567174061049

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