Properties

Degree 2
Conductor $ 2^{4} \cdot 3 $
Sign $-0.933 - 0.357i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.84 + 0.777i)2-s + (−1.22 + 1.22i)3-s + (2.79 − 2.86i)4-s + (−4.78 + 4.78i)5-s + (1.30 − 3.20i)6-s − 10.3·7-s + (−2.91 + 7.45i)8-s − 2.99i·9-s + (5.09 − 12.5i)10-s + (−0.526 − 0.526i)11-s + (0.0930 + 6.92i)12-s + (17.2 + 17.2i)13-s + (19.0 − 8.03i)14-s − 11.7i·15-s + (−0.429 − 15.9i)16-s + 4.71·17-s + ⋯
L(s)  = 1  + (−0.921 + 0.388i)2-s + (−0.408 + 0.408i)3-s + (0.697 − 0.716i)4-s + (−0.957 + 0.957i)5-s + (0.217 − 0.534i)6-s − 1.47·7-s + (−0.364 + 0.931i)8-s − 0.333i·9-s + (0.509 − 1.25i)10-s + (−0.0478 − 0.0478i)11-s + (0.00775 + 0.577i)12-s + (1.32 + 1.32i)13-s + (1.35 − 0.573i)14-s − 0.781i·15-s + (−0.0268 − 0.999i)16-s + 0.277·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(48\)    =    \(2^{4} \cdot 3\)
\( \varepsilon \)  =  $-0.933 - 0.357i$
motivic weight  =  \(2\)
character  :  $\chi_{48} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 48,\ (\ :1),\ -0.933 - 0.357i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0629719 + 0.340402i\)
\(L(\frac12)\)  \(\approx\)  \(0.0629719 + 0.340402i\)
\(L(2)\)   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 \( 1 + (1.84 - 0.777i)T \)
3 \( 1 + (1.22 - 1.22i)T \)
good5 \( 1 + (4.78 - 4.78i)T - 25iT^{2} \)
7 \( 1 + 10.3T + 49T^{2} \)
11 \( 1 + (0.526 + 0.526i)T + 121iT^{2} \)
13 \( 1 + (-17.2 - 17.2i)T + 169iT^{2} \)
17 \( 1 - 4.71T + 289T^{2} \)
19 \( 1 + (2.53 - 2.53i)T - 361iT^{2} \)
23 \( 1 + 12.5T + 529T^{2} \)
29 \( 1 + (2.19 + 2.19i)T + 841iT^{2} \)
31 \( 1 - 28.0iT - 961T^{2} \)
37 \( 1 + (32.1 - 32.1i)T - 1.36e3iT^{2} \)
41 \( 1 + 23.1iT - 1.68e3T^{2} \)
43 \( 1 + (-4.79 - 4.79i)T + 1.84e3iT^{2} \)
47 \( 1 + 39.0iT - 2.20e3T^{2} \)
53 \( 1 + (27.9 - 27.9i)T - 2.80e3iT^{2} \)
59 \( 1 + (-79.8 - 79.8i)T + 3.48e3iT^{2} \)
61 \( 1 + (36.7 + 36.7i)T + 3.72e3iT^{2} \)
67 \( 1 + (10.9 - 10.9i)T - 4.48e3iT^{2} \)
71 \( 1 - 52.6T + 5.04e3T^{2} \)
73 \( 1 + 67.8iT - 5.32e3T^{2} \)
79 \( 1 - 56.4iT - 6.24e3T^{2} \)
83 \( 1 + (58.3 - 58.3i)T - 6.88e3iT^{2} \)
89 \( 1 - 131. iT - 7.92e3T^{2} \)
97 \( 1 - 60.9T + 9.40e3T^{2} \)
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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

−16.00700335030649803580166157904, −15.35761418726261111186503233930, −13.96656529575482507407868602656, −12.00735153462705153357114514190, −10.99880154987976183564957314755, −10.01246272634389956576559926436, −8.734857522999047213730034458925, −7.02875048144783592790963465225, −6.22577571899850869857821276501, −3.54736934419977414921133792220, 0.49614870507904859657659736481, 3.54790274433153807732361457807, 6.12535704562413781966750438704, 7.69009386897323549059180946484, 8.738685452439597225442489337739, 10.12946265949644372215873927041, 11.40793802827014359211937876288, 12.62439923385948609598135588456, 13.02508946677102716289408745725, 15.78520197405580737027596421587

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