Properties

Label 2-48-48.29-c2-0-2
Degree $2$
Conductor $48$
Sign $0.969 - 0.244i$
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  + (−1.21 − 1.59i)2-s + (1.14 + 2.77i)3-s + (−1.06 + 3.85i)4-s + (4.80 + 4.80i)5-s + (3.01 − 5.18i)6-s − 7.36i·7-s + (7.42 − 2.97i)8-s + (−6.35 + 6.37i)9-s + (1.82 − 13.4i)10-s + (−0.514 − 0.514i)11-s + (−11.9 + 1.48i)12-s + (7.12 + 7.12i)13-s + (−11.7 + 8.91i)14-s + (−7.79 + 18.8i)15-s + (−13.7 − 8.21i)16-s − 11.1i·17-s + ⋯
L(s)  = 1  + (−0.605 − 0.795i)2-s + (0.383 + 0.923i)3-s + (−0.266 + 0.963i)4-s + (0.960 + 0.960i)5-s + (0.502 − 0.864i)6-s − 1.05i·7-s + (0.928 − 0.372i)8-s + (−0.706 + 0.707i)9-s + (0.182 − 1.34i)10-s + (−0.0467 − 0.0467i)11-s + (−0.992 + 0.123i)12-s + (0.548 + 0.548i)13-s + (−0.836 + 0.637i)14-s + (−0.519 + 1.25i)15-s + (−0.858 − 0.513i)16-s − 0.653i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.969 - 0.244i$
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: \((2,\ 48,\ (\ :1),\ 0.969 - 0.244i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.00907 + 0.125247i\)
\(L(\frac12)\) \(\approx\) \(1.00907 + 0.125247i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.21 + 1.59i)T \)
3 \( 1 + (-1.14 - 2.77i)T \)
good5 \( 1 + (-4.80 - 4.80i)T + 25iT^{2} \)
7 \( 1 + 7.36iT - 49T^{2} \)
11 \( 1 + (0.514 + 0.514i)T + 121iT^{2} \)
13 \( 1 + (-7.12 - 7.12i)T + 169iT^{2} \)
17 \( 1 + 11.1iT - 289T^{2} \)
19 \( 1 + (21.1 + 21.1i)T + 361iT^{2} \)
23 \( 1 + 7.80T + 529T^{2} \)
29 \( 1 + (-34.6 + 34.6i)T - 841iT^{2} \)
31 \( 1 - 24.8T + 961T^{2} \)
37 \( 1 + (18.2 - 18.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 64.2T + 1.68e3T^{2} \)
43 \( 1 + (-7.24 + 7.24i)T - 1.84e3iT^{2} \)
47 \( 1 - 23.0iT - 2.20e3T^{2} \)
53 \( 1 + (-31.9 - 31.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (-17.6 - 17.6i)T + 3.48e3iT^{2} \)
61 \( 1 + (12.3 + 12.3i)T + 3.72e3iT^{2} \)
67 \( 1 + (-41.1 - 41.1i)T + 4.48e3iT^{2} \)
71 \( 1 + 25.6T + 5.04e3T^{2} \)
73 \( 1 - 56.1iT - 5.32e3T^{2} \)
79 \( 1 + 35.7T + 6.24e3T^{2} \)
83 \( 1 + (94.9 - 94.9i)T - 6.88e3iT^{2} \)
89 \( 1 + 44.8T + 7.92e3T^{2} \)
97 \( 1 + 82.3T + 9.40e3T^{2} \)
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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

−15.56941331176165591556143169283, −13.93434908721069393131336600796, −13.58814604389011099627413946763, −11.42028305677996251165479200891, −10.45950113293250071941156585066, −9.864876012241251971467381772928, −8.528315472018901450610736037201, −6.79359807696952742886726454268, −4.29433517182403052146038597240, −2.65783786442539035316189742535, 1.72115708998705891254197700275, 5.49285891820605978936072386330, 6.39355983240617571828848412478, 8.354453200121895821728718400189, 8.755390433510863707735257593705, 10.20917685388190172663269464402, 12.27906696596698943629502406073, 13.19940787714631711110967301284, 14.30570795942934227705817558277, 15.40046589582275597844415319981

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