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Properties

Label 2-48-16.13-c1-0-1
Degree $2$
Conductor $48$
Sign $0.762 - 0.646i$
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  + (0.874 + 1.11i)2-s + (−0.707 − 0.707i)3-s + (−0.470 + 1.94i)4-s + (−0.334 + 0.334i)5-s + (0.167 − 1.40i)6-s − 4.55i·7-s + (−2.57 + 1.17i)8-s + 1.00i·9-s + (−0.665 − 0.0793i)10-s + (−2.47 + 2.47i)11-s + (1.70 − 1.04i)12-s + (−0.0594 − 0.0594i)13-s + (5.06 − 3.98i)14-s + 0.473·15-s + (−3.55 − 1.82i)16-s + 3.61·17-s + ⋯
L(s)  = 1  + (0.618 + 0.785i)2-s + (−0.408 − 0.408i)3-s + (−0.235 + 0.971i)4-s + (−0.149 + 0.149i)5-s + (0.0683 − 0.573i)6-s − 1.72i·7-s + (−0.909 + 0.416i)8-s + 0.333i·9-s + (−0.210 − 0.0250i)10-s + (−0.745 + 0.745i)11-s + (0.492 − 0.300i)12-s + (−0.0164 − 0.0164i)13-s + (1.35 − 1.06i)14-s + 0.122·15-s + (−0.889 − 0.457i)16-s + 0.877·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.762 - 0.646i$
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: \((2,\ 48,\ (\ :1/2),\ 0.762 - 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.859126 + 0.315241i\)
\(L(\frac12)\) \(\approx\) \(0.859126 + 0.315241i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.874 - 1.11i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + (0.334 - 0.334i)T - 5iT^{2} \)
7 \( 1 + 4.55iT - 7T^{2} \)
11 \( 1 + (2.47 - 2.47i)T - 11iT^{2} \)
13 \( 1 + (0.0594 + 0.0594i)T + 13iT^{2} \)
17 \( 1 - 3.61T + 17T^{2} \)
19 \( 1 + (-2.55 - 2.55i)T + 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (5.16 + 5.16i)T + 29iT^{2} \)
31 \( 1 + 0.557T + 31T^{2} \)
37 \( 1 + (-4.38 + 4.38i)T - 37iT^{2} \)
41 \( 1 + 9.27iT - 41T^{2} \)
43 \( 1 + (1.61 - 1.61i)T - 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (0.493 - 0.493i)T - 53iT^{2} \)
59 \( 1 + (-4 + 4i)T - 59iT^{2} \)
61 \( 1 + (-2.72 - 2.72i)T + 61iT^{2} \)
67 \( 1 + (-3.77 - 3.77i)T + 67iT^{2} \)
71 \( 1 - 9.11iT - 71T^{2} \)
73 \( 1 + 0.541iT - 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + (10.6 + 10.6i)T + 83iT^{2} \)
89 \( 1 - 14.6iT - 89T^{2} \)
97 \( 1 - 4.31T + 97T^{2} \)
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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

−15.89760953308572664773995942205, −14.57766008330093553152231520766, −13.57402103460834438797867056377, −12.73589645070448832286116986788, −11.36083120799465652053750283849, −9.954645857861525156894588031065, −7.66130997167112872618999970533, −7.25469706611507216003136633047, −5.50016234533184009125213007544, −3.88431800021340996070071180644, 2.93277313035644301932611142198, 5.04018784539916012716801928425, 5.97148374437644136007961728477, 8.596318950109452418931733201385, 9.800984711243321477277656143961, 11.16531769844445068984838941454, 12.03702935072545998173445842602, 12.97197468381363123603374713277, 14.48970710928985023142069963813, 15.45259046146594049541990349980

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