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Properties

Label 2-48-16.13-c1-0-3
Degree $2$
Conductor $48$
Sign $0.0819 + 0.996i$
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  + (−0.167 − 1.40i)2-s + (−0.707 − 0.707i)3-s + (−1.94 + 0.470i)4-s + (1.74 − 1.74i)5-s + (−0.874 + 1.11i)6-s + 2.55i·7-s + (0.985 + 2.65i)8-s + 1.00i·9-s + (−2.74 − 2.16i)10-s + (0.473 − 0.473i)11-s + (1.70 + 1.04i)12-s + (2.88 + 2.88i)13-s + (3.59 − 0.428i)14-s − 2.47·15-s + (3.55 − 1.82i)16-s − 6.44·17-s + ⋯
L(s)  = 1  + (−0.118 − 0.992i)2-s + (−0.408 − 0.408i)3-s + (−0.971 + 0.235i)4-s + (0.782 − 0.782i)5-s + (−0.357 + 0.453i)6-s + 0.966i·7-s + (0.348 + 0.937i)8-s + 0.333i·9-s + (−0.869 − 0.684i)10-s + (0.142 − 0.142i)11-s + (0.492 + 0.300i)12-s + (0.800 + 0.800i)13-s + (0.959 − 0.114i)14-s − 0.638·15-s + (0.889 − 0.457i)16-s − 1.56·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0819 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0819 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.0819 + 0.996i$
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: \((2,\ 48,\ (\ :1/2),\ 0.0819 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.523275 - 0.482015i\)
\(L(\frac12)\) \(\approx\) \(0.523275 - 0.482015i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.167 + 1.40i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + (-1.74 + 1.74i)T - 5iT^{2} \)
7 \( 1 - 2.55iT - 7T^{2} \)
11 \( 1 + (-0.473 + 0.473i)T - 11iT^{2} \)
13 \( 1 + (-2.88 - 2.88i)T + 13iT^{2} \)
17 \( 1 + 6.44T + 17T^{2} \)
19 \( 1 + (4.55 + 4.55i)T + 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (3.07 + 3.07i)T + 29iT^{2} \)
31 \( 1 - 6.55T + 31T^{2} \)
37 \( 1 + (2.72 - 2.72i)T - 37iT^{2} \)
41 \( 1 - 0.788iT - 41T^{2} \)
43 \( 1 + (0.389 - 0.389i)T - 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (2.57 - 2.57i)T - 53iT^{2} \)
59 \( 1 + (-4 + 4i)T - 59iT^{2} \)
61 \( 1 + (4.38 + 4.38i)T + 61iT^{2} \)
67 \( 1 + (2.11 + 2.11i)T + 67iT^{2} \)
71 \( 1 + 5.11iT - 71T^{2} \)
73 \( 1 + 14.7iT - 73T^{2} \)
79 \( 1 + 6.31T + 79T^{2} \)
83 \( 1 + (-0.641 - 0.641i)T + 83iT^{2} \)
89 \( 1 - 6.31iT - 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−15.48051233476787987913413189551, −13.62775374852836992565046773185, −13.10972206422193205943450347967, −11.89760806884588907529415522649, −10.97835760555487732646986991590, −9.276440395328109327772756639585, −8.649632203526046261570346763285, −6.20256932774519786608727753118, −4.70926013905730747314928032047, −2.03485356374126472316235332668, 4.17363002802444845124025890090, 5.97673243522385519688165266314, 6.90029031365375151745718942806, 8.585883936881300580487644998494, 10.16484221305086575658083319384, 10.73615605553485390077108513179, 12.96512922919880004350145842053, 13.94237730847152636270201588778, 14.92050384116219684512585704925, 15.99232959161946239588504434075

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