Properties

Label 2-48-4.3-c12-0-8
Degree $2$
Conductor $48$
Sign $0.866 + 0.5i$
Analytic cond. $43.8717$
Root an. cond. $6.62357$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 420. i·3-s + 1.03e4·5-s + 2.57e4i·7-s − 1.77e5·9-s − 2.20e6i·11-s + 5.65e6·13-s + 4.33e6i·15-s − 2.37e7·17-s − 5.50e7i·19-s − 1.08e7·21-s + 2.18e7i·23-s − 1.37e8·25-s − 7.45e7i·27-s + 3.88e8·29-s − 3.16e8i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.659·5-s + 0.218i·7-s − 0.333·9-s − 1.24i·11-s + 1.17·13-s + 0.380i·15-s − 0.982·17-s − 1.17i·19-s − 0.126·21-s + 0.147i·23-s − 0.564·25-s − 0.192i·27-s + 0.652·29-s − 0.357i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(43.8717\)
Root analytic conductor: \(6.62357\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :6),\ 0.866 + 0.5i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.171530493\)
\(L(\frac12)\) \(\approx\) \(2.171530493\)
\(L(7)\) 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 \( 1 - 420. iT \)
good5 \( 1 - 1.03e4T + 2.44e8T^{2} \)
7 \( 1 - 2.57e4iT - 1.38e10T^{2} \)
11 \( 1 + 2.20e6iT - 3.13e12T^{2} \)
13 \( 1 - 5.65e6T + 2.32e13T^{2} \)
17 \( 1 + 2.37e7T + 5.82e14T^{2} \)
19 \( 1 + 5.50e7iT - 2.21e15T^{2} \)
23 \( 1 - 2.18e7iT - 2.19e16T^{2} \)
29 \( 1 - 3.88e8T + 3.53e17T^{2} \)
31 \( 1 + 3.16e8iT - 7.87e17T^{2} \)
37 \( 1 - 1.17e9T + 6.58e18T^{2} \)
41 \( 1 - 3.32e9T + 2.25e19T^{2} \)
43 \( 1 + 6.04e9iT - 3.99e19T^{2} \)
47 \( 1 + 2.63e9iT - 1.16e20T^{2} \)
53 \( 1 - 2.86e10T + 4.91e20T^{2} \)
59 \( 1 + 6.66e10iT - 1.77e21T^{2} \)
61 \( 1 - 4.54e10T + 2.65e21T^{2} \)
67 \( 1 - 5.74e10iT - 8.18e21T^{2} \)
71 \( 1 + 2.96e9iT - 1.64e22T^{2} \)
73 \( 1 - 2.94e11T + 2.29e22T^{2} \)
79 \( 1 + 2.19e11iT - 5.90e22T^{2} \)
83 \( 1 + 3.91e10iT - 1.06e23T^{2} \)
89 \( 1 - 1.53e11T + 2.46e23T^{2} \)
97 \( 1 + 1.21e12T + 6.93e23T^{2} \)
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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

−13.28784111351990267532653675999, −11.48490579237573127286723676857, −10.68882337097021749241369831667, −9.278163756072993582234845795923, −8.449179953700750086459825559309, −6.46708496067575522072396924590, −5.44461989718939175091931234745, −3.87170926014751177166051518280, −2.43821137528930516635070170926, −0.66120921594699206878598908342, 1.20835996914797871039960536674, 2.29698603177724281578608353303, 4.15006887944541264230328198471, 5.82039906161990514585635480996, 6.89485868811338500152151196875, 8.249477848298314217456047707809, 9.596437896313194304799677005521, 10.77390942075497052135969200192, 12.16221749340390297180709514908, 13.19496029095920352578581283607

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