Properties

Label 2-48-48.29-c2-0-0
Degree $2$
Conductor $48$
Sign $0.434 - 0.900i$
Analytic cond. $1.30790$
Root an. cond. $1.14363$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.96 − 0.391i)2-s + (−2.99 − 0.164i)3-s + (3.69 + 1.53i)4-s + (3.61 + 3.61i)5-s + (5.81 + 1.49i)6-s + 12.2i·7-s + (−6.64 − 4.45i)8-s + (8.94 + 0.985i)9-s + (−5.67 − 8.49i)10-s + (−1.76 − 1.76i)11-s + (−10.8 − 5.20i)12-s + (−2.38 − 2.38i)13-s + (4.80 − 24.0i)14-s + (−10.2 − 11.4i)15-s + (11.2 + 11.3i)16-s + 20.0i·17-s + ⋯
L(s)  = 1  + (−0.980 − 0.195i)2-s + (−0.998 − 0.0548i)3-s + (0.923 + 0.383i)4-s + (0.722 + 0.722i)5-s + (0.968 + 0.249i)6-s + 1.75i·7-s + (−0.830 − 0.556i)8-s + (0.993 + 0.109i)9-s + (−0.567 − 0.849i)10-s + (−0.160 − 0.160i)11-s + (−0.901 − 0.433i)12-s + (−0.183 − 0.183i)13-s + (0.342 − 1.72i)14-s + (−0.681 − 0.761i)15-s + (0.705 + 0.708i)16-s + 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.434 - 0.900i$
Analytic conductor: \(1.30790\)
Root analytic conductor: \(1.14363\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1),\ 0.434 - 0.900i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.500463 + 0.314254i\)
\(L(\frac12)\) \(\approx\) \(0.500463 + 0.314254i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.96 + 0.391i)T \)
3 \( 1 + (2.99 + 0.164i)T \)
good5 \( 1 + (-3.61 - 3.61i)T + 25iT^{2} \)
7 \( 1 - 12.2iT - 49T^{2} \)
11 \( 1 + (1.76 + 1.76i)T + 121iT^{2} \)
13 \( 1 + (2.38 + 2.38i)T + 169iT^{2} \)
17 \( 1 - 20.0iT - 289T^{2} \)
19 \( 1 + (8.77 + 8.77i)T + 361iT^{2} \)
23 \( 1 - 13.1T + 529T^{2} \)
29 \( 1 + (-6.51 + 6.51i)T - 841iT^{2} \)
31 \( 1 - 37.5T + 961T^{2} \)
37 \( 1 + (-10.0 + 10.0i)T - 1.36e3iT^{2} \)
41 \( 1 - 4.57T + 1.68e3T^{2} \)
43 \( 1 + (-21.2 + 21.2i)T - 1.84e3iT^{2} \)
47 \( 1 + 54.8iT - 2.20e3T^{2} \)
53 \( 1 + (21.5 + 21.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (-53.6 - 53.6i)T + 3.48e3iT^{2} \)
61 \( 1 + (19.2 + 19.2i)T + 3.72e3iT^{2} \)
67 \( 1 + (-31.5 - 31.5i)T + 4.48e3iT^{2} \)
71 \( 1 - 65.1T + 5.04e3T^{2} \)
73 \( 1 - 50.2iT - 5.32e3T^{2} \)
79 \( 1 - 20.9T + 6.24e3T^{2} \)
83 \( 1 + (-6.35 + 6.35i)T - 6.88e3iT^{2} \)
89 \( 1 + 166.T + 7.92e3T^{2} \)
97 \( 1 - 139.T + 9.40e3T^{2} \)
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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

−15.74032757374721404861281057491, −14.99312451892882563244987140198, −12.85906135672923930109366821485, −11.89799229998788322114685465807, −10.81534459639346283155093840421, −9.826375763296552768656480833700, −8.472668645524662443307506753323, −6.61924933976225630628313164064, −5.72661164439818116645891503875, −2.35795213908992134827978499804, 1.00129983075346991908617434918, 4.84693040670722903400486027896, 6.48942570030888008556461947424, 7.60073362345779973750566820125, 9.491290665251720255445204759199, 10.31554884407511534229352898564, 11.34491455497807255671744765441, 12.80344978184280210142891910242, 14.06140701813957992470808886869, 15.82806625545191367712600499558

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