Properties

Label 2-48-16.13-c5-0-12
Degree $2$
Conductor $48$
Sign $-0.597 + 0.801i$
Analytic cond. $7.69842$
Root an. cond. $2.77460$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.74 − 4.24i)2-s + (6.36 + 6.36i)3-s + (−3.95 + 31.7i)4-s + (−41.3 + 41.3i)5-s + (3.15 − 50.8i)6-s − 215. i·7-s + (149. − 102. i)8-s + 81i·9-s + (329. + 20.4i)10-s + (275. − 275. i)11-s + (−227. + 176. i)12-s + (−728. − 728. i)13-s + (−913. + 806. i)14-s − 525.·15-s + (−992. − 251. i)16-s + 1.32e3·17-s + ⋯
L(s)  = 1  + (−0.661 − 0.749i)2-s + (0.408 + 0.408i)3-s + (−0.123 + 0.992i)4-s + (−0.739 + 0.739i)5-s + (0.0357 − 0.576i)6-s − 1.66i·7-s + (0.825 − 0.564i)8-s + 0.333i·9-s + (1.04 + 0.0647i)10-s + (0.687 − 0.687i)11-s + (−0.455 + 0.354i)12-s + (−1.19 − 1.19i)13-s + (−1.24 + 1.10i)14-s − 0.603·15-s + (−0.969 − 0.245i)16-s + 1.11·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.597 + 0.801i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ -0.597 + 0.801i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.358068 - 0.713648i\)
\(L(\frac12)\) \(\approx\) \(0.358068 - 0.713648i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.74 + 4.24i)T \)
3 \( 1 + (-6.36 - 6.36i)T \)
good5 \( 1 + (41.3 - 41.3i)T - 3.12e3iT^{2} \)
7 \( 1 + 215. iT - 1.68e4T^{2} \)
11 \( 1 + (-275. + 275. i)T - 1.61e5iT^{2} \)
13 \( 1 + (728. + 728. i)T + 3.71e5iT^{2} \)
17 \( 1 - 1.32e3T + 1.41e6T^{2} \)
19 \( 1 + (1.67e3 + 1.67e3i)T + 2.47e6iT^{2} \)
23 \( 1 - 1.36e3iT - 6.43e6T^{2} \)
29 \( 1 + (4.95e3 + 4.95e3i)T + 2.05e7iT^{2} \)
31 \( 1 - 6.20e3T + 2.86e7T^{2} \)
37 \( 1 + (-4.05e3 + 4.05e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 4.42e3iT - 1.15e8T^{2} \)
43 \( 1 + (-968. + 968. i)T - 1.47e8iT^{2} \)
47 \( 1 + 7.74e3T + 2.29e8T^{2} \)
53 \( 1 + (7.48e3 - 7.48e3i)T - 4.18e8iT^{2} \)
59 \( 1 + (9.19e3 - 9.19e3i)T - 7.14e8iT^{2} \)
61 \( 1 + (1.83e4 + 1.83e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (-2.07e4 - 2.07e4i)T + 1.35e9iT^{2} \)
71 \( 1 - 4.22e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.87e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.53e4T + 3.07e9T^{2} \)
83 \( 1 + (-1.80e4 - 1.80e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 2.11e4iT - 5.58e9T^{2} \)
97 \( 1 - 8.57e4T + 8.58e9T^{2} \)
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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

−14.20631635076552841126695314103, −13.03146426858510849652479977070, −11.46647848708855076226461491870, −10.60819300760428293437313905499, −9.720772903661276914725843239297, −7.983970025033820665337853481989, −7.23182398855135002958857267692, −4.14875242014814519568932073112, −3.09804759883997306326988154022, −0.49534405953500056728701276028, 1.86092836330510412142161747618, 4.73822501489064724164700843604, 6.36192640348497474419929117946, 7.83731334506587807822633557153, 8.802123295248987327024390677130, 9.670848473852151214842520381315, 11.90882981870723684721300048371, 12.45145470886697977633481047400, 14.56066400600424022437924299400, 14.90992187717370098448707679554

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