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Properties

Label 2-48-48.11-c3-0-12
Degree $2$
Conductor $48$
Sign $0.213 - 0.976i$
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  + (1.70 + 2.25i)2-s + (3.47 + 3.86i)3-s + (−2.19 + 7.69i)4-s + (13.5 − 13.5i)5-s + (−2.79 + 14.4i)6-s − 19.7·7-s + (−21.1 + 8.14i)8-s + (−2.80 + 26.8i)9-s + (53.7 + 7.52i)10-s + (−20.5 − 20.5i)11-s + (−37.3 + 18.2i)12-s + (36.7 − 36.7i)13-s + (−33.6 − 44.5i)14-s + (99.6 + 5.19i)15-s + (−54.3 − 33.7i)16-s + 4.20i·17-s + ⋯
L(s)  = 1  + (0.602 + 0.798i)2-s + (0.669 + 0.742i)3-s + (−0.274 + 0.961i)4-s + (1.21 − 1.21i)5-s + (−0.189 + 0.981i)6-s − 1.06·7-s + (−0.932 + 0.360i)8-s + (−0.103 + 0.994i)9-s + (1.70 + 0.238i)10-s + (−0.562 − 0.562i)11-s + (−0.898 + 0.439i)12-s + (0.783 − 0.783i)13-s + (−0.641 − 0.850i)14-s + (1.71 + 0.0893i)15-s + (−0.849 − 0.527i)16-s + 0.0599i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.213 - 0.976i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ 0.213 - 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.68888 + 1.35920i\)
\(L(\frac12)\) \(\approx\) \(1.68888 + 1.35920i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.70 - 2.25i)T \)
3 \( 1 + (-3.47 - 3.86i)T \)
good5 \( 1 + (-13.5 + 13.5i)T - 125iT^{2} \)
7 \( 1 + 19.7T + 343T^{2} \)
11 \( 1 + (20.5 + 20.5i)T + 1.33e3iT^{2} \)
13 \( 1 + (-36.7 + 36.7i)T - 2.19e3iT^{2} \)
17 \( 1 - 4.20iT - 4.91e3T^{2} \)
19 \( 1 + (-38.6 - 38.6i)T + 6.85e3iT^{2} \)
23 \( 1 + 69.4iT - 1.21e4T^{2} \)
29 \( 1 + (-23.0 - 23.0i)T + 2.43e4iT^{2} \)
31 \( 1 - 219. iT - 2.97e4T^{2} \)
37 \( 1 + (-68.0 - 68.0i)T + 5.06e4iT^{2} \)
41 \( 1 + 325.T + 6.89e4T^{2} \)
43 \( 1 + (36.9 - 36.9i)T - 7.95e4iT^{2} \)
47 \( 1 - 192.T + 1.03e5T^{2} \)
53 \( 1 + (461. - 461. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-0.977 - 0.977i)T + 2.05e5iT^{2} \)
61 \( 1 + (-216. + 216. i)T - 2.26e5iT^{2} \)
67 \( 1 + (27.8 + 27.8i)T + 3.00e5iT^{2} \)
71 \( 1 + 786. iT - 3.57e5T^{2} \)
73 \( 1 - 510. iT - 3.89e5T^{2} \)
79 \( 1 + 230. iT - 4.93e5T^{2} \)
83 \( 1 + (593. - 593. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 - 805.T + 9.12e5T^{2} \)
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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

−15.67440257654260118063085657692, −14.09264979438603437708461823114, −13.35091049489235727801373307593, −12.63835351096647694316079046373, −10.27459101093655211506541725291, −9.117115072759212015187772568349, −8.253480531490727185159025631935, −6.07446052901745035563885202192, −5.02738482493173384491418579395, −3.20323922164531164844310988636, 2.11446048703646519591140489125, 3.30738364416842769015864103136, 6.01205213234308186057388402540, 6.92206540003024058224605305767, 9.333738678478752374790830030249, 10.06065794490514225463480666393, 11.50001258859184883722170371680, 13.07063679274393896225059868678, 13.52528770036798990833239449881, 14.45634903275900202311103730419

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