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Properties

Label 2-48-48.11-c3-0-1
Degree $2$
Conductor $48$
Sign $-0.950 + 0.310i$
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  + (−2.81 − 0.221i)2-s + (−2.71 + 4.43i)3-s + (7.90 + 1.24i)4-s + (3.17 − 3.17i)5-s + (8.62 − 11.8i)6-s − 32.3·7-s + (−22.0 − 5.26i)8-s + (−12.2 − 24.0i)9-s + (−9.65 + 8.25i)10-s + (−16.0 − 16.0i)11-s + (−26.9 + 31.6i)12-s + (−18.2 + 18.2i)13-s + (91.2 + 7.15i)14-s + (5.45 + 22.6i)15-s + (60.8 + 19.7i)16-s − 38.5i·17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0781i)2-s + (−0.522 + 0.852i)3-s + (0.987 + 0.155i)4-s + (0.284 − 0.284i)5-s + (0.587 − 0.809i)6-s − 1.74·7-s + (−0.972 − 0.232i)8-s + (−0.454 − 0.890i)9-s + (−0.305 + 0.260i)10-s + (−0.441 − 0.441i)11-s + (−0.648 + 0.761i)12-s + (−0.390 + 0.390i)13-s + (1.74 + 0.136i)14-s + (0.0939 + 0.390i)15-s + (0.951 + 0.307i)16-s − 0.550i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.310i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.950 + 0.310i$
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: \((2,\ 48,\ (\ :3/2),\ -0.950 + 0.310i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00539305 - 0.0338563i\)
\(L(\frac12)\) \(\approx\) \(0.00539305 - 0.0338563i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.81 + 0.221i)T \)
3 \( 1 + (2.71 - 4.43i)T \)
good5 \( 1 + (-3.17 + 3.17i)T - 125iT^{2} \)
7 \( 1 + 32.3T + 343T^{2} \)
11 \( 1 + (16.0 + 16.0i)T + 1.33e3iT^{2} \)
13 \( 1 + (18.2 - 18.2i)T - 2.19e3iT^{2} \)
17 \( 1 + 38.5iT - 4.91e3T^{2} \)
19 \( 1 + (56.2 + 56.2i)T + 6.85e3iT^{2} \)
23 \( 1 - 197. iT - 1.21e4T^{2} \)
29 \( 1 + (57.3 + 57.3i)T + 2.43e4iT^{2} \)
31 \( 1 - 148. iT - 2.97e4T^{2} \)
37 \( 1 + (72.5 + 72.5i)T + 5.06e4iT^{2} \)
41 \( 1 + 73.1T + 6.89e4T^{2} \)
43 \( 1 + (-226. + 226. i)T - 7.95e4iT^{2} \)
47 \( 1 + 412.T + 1.03e5T^{2} \)
53 \( 1 + (94.8 - 94.8i)T - 1.48e5iT^{2} \)
59 \( 1 + (-344. - 344. i)T + 2.05e5iT^{2} \)
61 \( 1 + (153. - 153. i)T - 2.26e5iT^{2} \)
67 \( 1 + (603. + 603. i)T + 3.00e5iT^{2} \)
71 \( 1 + 711. iT - 3.57e5T^{2} \)
73 \( 1 - 687. iT - 3.89e5T^{2} \)
79 \( 1 + 162. iT - 4.93e5T^{2} \)
83 \( 1 + (-748. + 748. i)T - 5.71e5iT^{2} \)
89 \( 1 + 927.T + 7.04e5T^{2} \)
97 \( 1 + 208.T + 9.12e5T^{2} \)
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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

−16.04487430366230753925350941620, −15.31255684708345314715139751137, −13.29651284436247881038186124328, −12.01461119733257432752327319550, −10.77682448111263750330530104920, −9.642964607296131385078789734599, −9.106001333311242492810839139728, −6.98129993837666727075558893531, −5.69643706341331155612298738500, −3.25271226189461231892171738873, 0.03501360954245075637526804263, 2.53181279953906394384892252502, 6.07806357898914277966445450531, 6.79929923282597279087484508828, 8.220685710056202859762937659944, 9.880246922747841300363726011165, 10.64012427636993124661031917121, 12.34783384565369179559512806790, 12.93336501329190211979835223043, 14.71887232371476609956464702179

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