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Properties

Label 2-48-16.5-c5-0-5
Degree $2$
Conductor $48$
Sign $0.995 - 0.0897i$
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  + (−1.76 − 5.37i)2-s + (6.36 − 6.36i)3-s + (−25.7 + 18.9i)4-s + (47.6 + 47.6i)5-s + (−45.4 − 22.9i)6-s + 212. i·7-s + (147. + 105. i)8-s − 81i·9-s + (171. − 339. i)10-s + (62.9 + 62.9i)11-s + (−43.4 + 284. i)12-s + (−312. + 312. i)13-s + (1.14e3 − 375. i)14-s + 606.·15-s + (305. − 977. i)16-s − 109.·17-s + ⋯
L(s)  = 1  + (−0.311 − 0.950i)2-s + (0.408 − 0.408i)3-s + (−0.805 + 0.592i)4-s + (0.851 + 0.851i)5-s + (−0.515 − 0.260i)6-s + 1.64i·7-s + (0.813 + 0.580i)8-s − 0.333i·9-s + (0.543 − 1.07i)10-s + (0.156 + 0.156i)11-s + (−0.0870 + 0.570i)12-s + (−0.512 + 0.512i)13-s + (1.55 − 0.511i)14-s + 0.695·15-s + (0.298 − 0.954i)16-s − 0.0921·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0897i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.995 - 0.0897i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ 0.995 - 0.0897i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.60707 + 0.0722621i\)
\(L(\frac12)\) \(\approx\) \(1.60707 + 0.0722621i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.76 + 5.37i)T \)
3 \( 1 + (-6.36 + 6.36i)T \)
good5 \( 1 + (-47.6 - 47.6i)T + 3.12e3iT^{2} \)
7 \( 1 - 212. iT - 1.68e4T^{2} \)
11 \( 1 + (-62.9 - 62.9i)T + 1.61e5iT^{2} \)
13 \( 1 + (312. - 312. i)T - 3.71e5iT^{2} \)
17 \( 1 + 109.T + 1.41e6T^{2} \)
19 \( 1 + (-2.14e3 + 2.14e3i)T - 2.47e6iT^{2} \)
23 \( 1 - 4.82e3iT - 6.43e6T^{2} \)
29 \( 1 + (833. - 833. i)T - 2.05e7iT^{2} \)
31 \( 1 + 1.51e3T + 2.86e7T^{2} \)
37 \( 1 + (4.03e3 + 4.03e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 4.21e3iT - 1.15e8T^{2} \)
43 \( 1 + (-8.42e3 - 8.42e3i)T + 1.47e8iT^{2} \)
47 \( 1 - 2.29e4T + 2.29e8T^{2} \)
53 \( 1 + (5.88e3 + 5.88e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (-1.64e4 - 1.64e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (-2.79e4 + 2.79e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (1.65e3 - 1.65e3i)T - 1.35e9iT^{2} \)
71 \( 1 + 4.87e4iT - 1.80e9T^{2} \)
73 \( 1 + 7.69e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.64e4T + 3.07e9T^{2} \)
83 \( 1 + (4.64e4 - 4.64e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 3.35e4iT - 5.58e9T^{2} \)
97 \( 1 + 5.00e4T + 8.58e9T^{2} \)
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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

−14.34694601240860151139098040736, −13.45192394013065260077118590515, −12.17879284765202513372696815000, −11.29005090658399625557175988309, −9.599503345382228516020225275429, −9.069003821367639179801303767939, −7.29170850264760580799278639888, −5.47920954097778807912918213716, −2.96516072433141718469986534235, −1.97450324474462652087303404609, 0.971389700159009171376288391958, 4.17426773021623052222288317002, 5.51732843930903225292880526167, 7.21941864918668586094774018481, 8.431528770218710656161916759714, 9.758839791414266531623779405660, 10.40207492724808698246183585503, 12.82619370921116063539286810226, 13.87622415610836768810700416690, 14.44860418459154504907862698718

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