Properties

Label 2-48-3.2-c10-0-15
Degree $2$
Conductor $48$
Sign $0.824 + 0.566i$
Analytic cond. $30.4971$
Root an. cond. $5.52242$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (200. + 137. i)3-s − 3.63e3i·5-s + 2.32e4·7-s + (2.11e4 + 5.51e4i)9-s − 6.24e4i·11-s − 1.70e5·13-s + (4.99e5 − 7.27e5i)15-s − 2.66e6i·17-s − 7.66e5·19-s + (4.65e6 + 3.19e6i)21-s − 1.40e6i·23-s − 3.41e6·25-s + (−3.34e6 + 1.39e7i)27-s + 4.83e6i·29-s + 4.18e7·31-s + ⋯
L(s)  = 1  + (0.824 + 0.566i)3-s − 1.16i·5-s + 1.38·7-s + (0.358 + 0.933i)9-s − 0.387i·11-s − 0.458·13-s + (0.657 − 0.957i)15-s − 1.87i·17-s − 0.309·19-s + (1.13 + 0.782i)21-s − 0.218i·23-s − 0.349·25-s + (−0.233 + 0.972i)27-s + 0.235i·29-s + 1.46·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.824 + 0.566i$
Analytic conductor: \(30.4971\)
Root analytic conductor: \(5.52242\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5),\ 0.824 + 0.566i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.92954 - 0.909572i\)
\(L(\frac12)\) \(\approx\) \(2.92954 - 0.909572i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-200. - 137. i)T \)
good5 \( 1 + 3.63e3iT - 9.76e6T^{2} \)
7 \( 1 - 2.32e4T + 2.82e8T^{2} \)
11 \( 1 + 6.24e4iT - 2.59e10T^{2} \)
13 \( 1 + 1.70e5T + 1.37e11T^{2} \)
17 \( 1 + 2.66e6iT - 2.01e12T^{2} \)
19 \( 1 + 7.66e5T + 6.13e12T^{2} \)
23 \( 1 + 1.40e6iT - 4.14e13T^{2} \)
29 \( 1 - 4.83e6iT - 4.20e14T^{2} \)
31 \( 1 - 4.18e7T + 8.19e14T^{2} \)
37 \( 1 - 5.01e7T + 4.80e15T^{2} \)
41 \( 1 + 1.49e8iT - 1.34e16T^{2} \)
43 \( 1 - 1.98e8T + 2.16e16T^{2} \)
47 \( 1 - 1.55e8iT - 5.25e16T^{2} \)
53 \( 1 - 4.21e7iT - 1.74e17T^{2} \)
59 \( 1 + 2.92e8iT - 5.11e17T^{2} \)
61 \( 1 + 5.30e8T + 7.13e17T^{2} \)
67 \( 1 + 5.22e8T + 1.82e18T^{2} \)
71 \( 1 + 5.71e8iT - 3.25e18T^{2} \)
73 \( 1 - 2.18e9T + 4.29e18T^{2} \)
79 \( 1 + 1.96e9T + 9.46e18T^{2} \)
83 \( 1 - 2.18e9iT - 1.55e19T^{2} \)
89 \( 1 + 2.38e8iT - 3.11e19T^{2} \)
97 \( 1 + 8.84e9T + 7.37e19T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.64969923929664653035386881591, −12.20547865257437286218362520636, −10.98506175756522784646203173503, −9.472846530740962931508311947544, −8.573685064173059397470926279384, −7.61621295251844831608711711442, −5.12788621826835726124693490939, −4.45109609705176941211543384423, −2.50109645652302692062248814703, −0.953873223108240585981725642596, 1.51502308893242200633719031819, 2.63401499170346589744028010827, 4.23399236582004066704141522009, 6.30541573102605911872587130057, 7.56583269676154654047569938387, 8.402714793225725746142988820001, 10.07998021395726622268800802634, 11.22127531117165443829325885170, 12.49295676385987939876530595553, 13.83753787493901251892927287902

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