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Properties

Label 2-48-16.13-c1-0-2
Degree $2$
Conductor $48$
Sign $0.773 + 0.633i$
Analytic cond. $0.383281$
Root an. cond. $0.619097$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.635 − 1.26i)2-s + (0.707 + 0.707i)3-s + (−1.19 − 1.60i)4-s + (−2.68 + 2.68i)5-s + (1.34 − 0.443i)6-s − 2.15i·7-s + (−2.78 + 0.484i)8-s + 1.00i·9-s + (1.68 + 5.09i)10-s + (1.79 − 1.79i)11-s + (0.292 − 1.97i)12-s + (1.38 + 1.38i)13-s + (−2.72 − 1.37i)14-s − 3.79·15-s + (−1.15 + 3.82i)16-s − 0.224·17-s + ⋯
L(s)  = 1  + (0.449 − 0.893i)2-s + (0.408 + 0.408i)3-s + (−0.595 − 0.803i)4-s + (−1.20 + 1.20i)5-s + (0.548 − 0.181i)6-s − 0.816i·7-s + (−0.985 + 0.171i)8-s + 0.333i·9-s + (0.533 + 1.61i)10-s + (0.542 − 0.542i)11-s + (0.0845 − 0.571i)12-s + (0.383 + 0.383i)13-s + (−0.728 − 0.366i)14-s − 0.980·15-s + (−0.289 + 0.957i)16-s − 0.0545·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.773 + 0.633i$
Analytic conductor: \(0.383281\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1/2),\ 0.773 + 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.871540 - 0.311560i\)
\(L(\frac12)\) \(\approx\) \(0.871540 - 0.311560i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.635 + 1.26i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
good5 \( 1 + (2.68 - 2.68i)T - 5iT^{2} \)
7 \( 1 + 2.15iT - 7T^{2} \)
11 \( 1 + (-1.79 + 1.79i)T - 11iT^{2} \)
13 \( 1 + (-1.38 - 1.38i)T + 13iT^{2} \)
17 \( 1 + 0.224T + 17T^{2} \)
19 \( 1 + (-0.158 - 0.158i)T + 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (1.85 + 1.85i)T + 29iT^{2} \)
31 \( 1 - 1.84T + 31T^{2} \)
37 \( 1 + (3.66 - 3.66i)T - 37iT^{2} \)
41 \( 1 - 5.88iT - 41T^{2} \)
43 \( 1 + (7.75 - 7.75i)T - 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (-7.51 + 7.51i)T - 53iT^{2} \)
59 \( 1 + (-4 + 4i)T - 59iT^{2} \)
61 \( 1 + (-5.98 - 5.98i)T + 61iT^{2} \)
67 \( 1 + (10.4 + 10.4i)T + 67iT^{2} \)
71 \( 1 - 4.31iT - 71T^{2} \)
73 \( 1 - 5.97iT - 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 83iT^{2} \)
89 \( 1 - 1.42iT - 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−15.15685271547203927047634424941, −14.41983232478527245787039181785, −13.45950518284363879693775285106, −11.73266118695565942172873152568, −11.02654065873239224617703655017, −10.01882682873606116854011464606, −8.340574390110295203834746480786, −6.67926877056311355946931200854, −4.23338644021661599711126263758, −3.26001583815327479995642235752, 3.86179068547023561844984227152, 5.37842133309248723250807203915, 7.20809824324198859285104900790, 8.381293715168107984791390773270, 9.086993693289973544211052872954, 11.88643337552233925390382200980, 12.43031386523642399712733001277, 13.52401554393183182785110398502, 15.02615018984691346279224474452, 15.60478899011562921687979644080

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