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Properties

Label 2-48-16.5-c3-0-0
Degree $2$
Conductor $48$
Sign $-0.590 + 0.807i$
Analytic cond. $2.83209$
Root an. cond. $1.68288$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.716 + 2.73i)2-s + (−2.12 + 2.12i)3-s + (−6.97 − 3.91i)4-s + (−11.7 − 11.7i)5-s + (−4.28 − 7.32i)6-s + 14.7i·7-s + (15.7 − 16.2i)8-s − 8.99i·9-s + (40.6 − 23.7i)10-s + (−24.9 − 24.9i)11-s + (23.1 − 6.48i)12-s + (−58.1 + 58.1i)13-s + (−40.2 − 10.5i)14-s + 49.9·15-s + (33.2 + 54.6i)16-s − 75.8·17-s + ⋯
L(s)  = 1  + (−0.253 + 0.967i)2-s + (−0.408 + 0.408i)3-s + (−0.871 − 0.489i)4-s + (−1.05 − 1.05i)5-s + (−0.291 − 0.498i)6-s + 0.794i·7-s + (0.694 − 0.719i)8-s − 0.333i·9-s + (1.28 − 0.752i)10-s + (−0.683 − 0.683i)11-s + (0.555 − 0.155i)12-s + (−1.24 + 1.24i)13-s + (−0.768 − 0.201i)14-s + 0.859·15-s + (0.520 + 0.854i)16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.590 + 0.807i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ -0.590 + 0.807i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0256628 - 0.0505471i\)
\(L(\frac12)\) \(\approx\) \(0.0256628 - 0.0505471i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.716 - 2.73i)T \)
3 \( 1 + (2.12 - 2.12i)T \)
good5 \( 1 + (11.7 + 11.7i)T + 125iT^{2} \)
7 \( 1 - 14.7iT - 343T^{2} \)
11 \( 1 + (24.9 + 24.9i)T + 1.33e3iT^{2} \)
13 \( 1 + (58.1 - 58.1i)T - 2.19e3iT^{2} \)
17 \( 1 + 75.8T + 4.91e3T^{2} \)
19 \( 1 + (-51.8 + 51.8i)T - 6.85e3iT^{2} \)
23 \( 1 - 149. iT - 1.21e4T^{2} \)
29 \( 1 + (-48.5 + 48.5i)T - 2.43e4iT^{2} \)
31 \( 1 - 29.6T + 2.97e4T^{2} \)
37 \( 1 + (147. + 147. i)T + 5.06e4iT^{2} \)
41 \( 1 + 225. iT - 6.89e4T^{2} \)
43 \( 1 + (-81.7 - 81.7i)T + 7.95e4iT^{2} \)
47 \( 1 - 46.5T + 1.03e5T^{2} \)
53 \( 1 + (156. + 156. i)T + 1.48e5iT^{2} \)
59 \( 1 + (238. + 238. i)T + 2.05e5iT^{2} \)
61 \( 1 + (594. - 594. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-299. + 299. i)T - 3.00e5iT^{2} \)
71 \( 1 - 693. iT - 3.57e5T^{2} \)
73 \( 1 + 462. iT - 3.89e5T^{2} \)
79 \( 1 + 878.T + 4.93e5T^{2} \)
83 \( 1 + (926. - 926. i)T - 5.71e5iT^{2} \)
89 \( 1 - 350. iT - 7.04e5T^{2} \)
97 \( 1 + 766.T + 9.12e5T^{2} \)
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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

−15.78568737459785498753276859233, −15.41620500038358735792904834950, −13.80116372528245032866583258890, −12.40378965866671867284434837568, −11.34246196989810705389018560600, −9.445749187331985245157746661587, −8.637749903840654419694713941531, −7.26743423790744790223624534254, −5.44685432559946712833653846197, −4.43361508513333143903505889385, 0.04853387784131095006470268946, 2.85356062811153844253354885204, 4.61476220164651451810066807650, 7.16786162355358710840941357383, 7.972894270349958105241811116715, 10.20589267745400185819943410827, 10.76051870129870626346559702452, 12.00023175634577708590203960667, 12.87870727727977483973034816276, 14.27683737011887246899206991563

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