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Properties

Label 2-48-48.35-c3-0-14
Degree $2$
Conductor $48$
Sign $0.111 + 0.993i$
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 + (3.86 − 3.47i)3-s + (−2.19 − 7.69i)4-s + (−13.5 − 13.5i)5-s + (1.27 + 14.6i)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 + (−35.2 − 22.0i)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 + (0.742 − 0.669i)3-s + (−0.274 − 0.961i)4-s + (−1.21 − 1.21i)5-s + (0.0868 + 0.996i)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 + (−0.847 − 0.530i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.111 + 0.993i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ 0.111 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.615522 - 0.550507i\)
\(L(\frac12)\) \(\approx\) \(0.615522 - 0.550507i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.86 + 3.47i)T \)
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

−15.18036004862406583336330752321, −13.78681553613599102508041325391, −12.82287167619856717217437945970, −11.51341194191391939452278663551, −9.311657116561932115386732995179, −8.720513620590267450459109769125, −7.57043996112636832060797400702, −6.27294838704499836768239416371, −4.02107790482526373328137834442, −0.75028334807162743170730507797, 3.08399852654346852131349676802, 3.82936744973260962485388808328, 7.09391421501882621586174162977, 8.232764649513817741730540688531, 9.654086517756231731218794028683, 10.54917833049926492937717929334, 11.61536378058275378610517720851, 12.98700949369745058546770514978, 14.39159346433814945150015685739, 15.59819843582563273538395161377

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