Properties

Label 2-48-16.5-c3-0-3
Degree $2$
Conductor $48$
Sign $-0.0810 - 0.996i$
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  + (−0.220 + 2.81i)2-s + (2.12 − 2.12i)3-s + (−7.90 − 1.24i)4-s + (10.2 + 10.2i)5-s + (5.51 + 6.44i)6-s + 32.8i·7-s + (5.23 − 22.0i)8-s − 8.99i·9-s + (−31.2 + 26.7i)10-s + (−18.2 − 18.2i)11-s + (−19.3 + 14.1i)12-s + (22.5 − 22.5i)13-s + (−92.5 − 7.22i)14-s + 43.6·15-s + (60.9 + 19.6i)16-s + 50.1·17-s + ⋯
L(s)  = 1  + (−0.0778 + 0.996i)2-s + (0.408 − 0.408i)3-s + (−0.987 − 0.155i)4-s + (0.920 + 0.920i)5-s + (0.375 + 0.438i)6-s + 1.77i·7-s + (0.231 − 0.972i)8-s − 0.333i·9-s + (−0.989 + 0.846i)10-s + (−0.499 − 0.499i)11-s + (−0.466 + 0.339i)12-s + (0.481 − 0.481i)13-s + (−1.76 − 0.137i)14-s + 0.751·15-s + (0.951 + 0.306i)16-s + 0.715·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0810 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0810 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.0810 - 0.996i$
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: \((2,\ 48,\ (\ :3/2),\ -0.0810 - 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.999166 + 1.08372i\)
\(L(\frac12)\) \(\approx\) \(0.999166 + 1.08372i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.220 - 2.81i)T \)
3 \( 1 + (-2.12 + 2.12i)T \)
good5 \( 1 + (-10.2 - 10.2i)T + 125iT^{2} \)
7 \( 1 - 32.8iT - 343T^{2} \)
11 \( 1 + (18.2 + 18.2i)T + 1.33e3iT^{2} \)
13 \( 1 + (-22.5 + 22.5i)T - 2.19e3iT^{2} \)
17 \( 1 - 50.1T + 4.91e3T^{2} \)
19 \( 1 + (6.68 - 6.68i)T - 6.85e3iT^{2} \)
23 \( 1 + 186. iT - 1.21e4T^{2} \)
29 \( 1 + (-118. + 118. i)T - 2.43e4iT^{2} \)
31 \( 1 + 250.T + 2.97e4T^{2} \)
37 \( 1 + (-198. - 198. i)T + 5.06e4iT^{2} \)
41 \( 1 + 186. iT - 6.89e4T^{2} \)
43 \( 1 + (-10.9 - 10.9i)T + 7.95e4iT^{2} \)
47 \( 1 - 23.1T + 1.03e5T^{2} \)
53 \( 1 + (-134. - 134. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-220. - 220. i)T + 2.05e5iT^{2} \)
61 \( 1 + (453. - 453. i)T - 2.26e5iT^{2} \)
67 \( 1 + (184. - 184. i)T - 3.00e5iT^{2} \)
71 \( 1 + 18.8iT - 3.57e5T^{2} \)
73 \( 1 + 828. iT - 3.89e5T^{2} \)
79 \( 1 - 1.04e3T + 4.93e5T^{2} \)
83 \( 1 + (173. - 173. i)T - 5.71e5iT^{2} \)
89 \( 1 - 335. iT - 7.04e5T^{2} \)
97 \( 1 + 687.T + 9.12e5T^{2} \)
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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

−15.20531585820277601577129795084, −14.56025090402860941377026957795, −13.48548714863013945609730508704, −12.36507137603392195714269774051, −10.40144396598291490132292418731, −9.083069026690404763087277667031, −8.087954917014549783947819040509, −6.37598493917465866386744516693, −5.62161551409682849707161192927, −2.71467913881193895021311253275, 1.41694261575109380428259167911, 3.78709459565377448655179353646, 5.12004266927910963320030664770, 7.69014491062083425677464836750, 9.245572766124051035240727178507, 10.01556659496407529057628612427, 11.05468715295429046018661938776, 12.82624265311239061564131100015, 13.53080455617173936553280667460, 14.33151620954949190536614801425

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