Properties

Label 2-48-4.3-c12-0-2
Degree $2$
Conductor $48$
Sign $0.866 - 0.5i$
Analytic cond. $43.8717$
Root an. cond. $6.62357$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 420. i·3-s − 2.12e4·5-s + 7.11e3i·7-s − 1.77e5·9-s − 1.88e6i·11-s − 8.56e6·13-s + 8.95e6i·15-s + 6.77e6·17-s + 6.45e5i·19-s + 2.99e6·21-s + 2.15e8i·23-s + 2.08e8·25-s + 7.45e7i·27-s − 3.66e8·29-s − 1.03e9i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.36·5-s + 0.0604i·7-s − 0.333·9-s − 1.06i·11-s − 1.77·13-s + 0.786i·15-s + 0.280·17-s + 0.0137i·19-s + 0.0349·21-s + 1.45i·23-s + 0.856·25-s + 0.192i·27-s − 0.616·29-s − 1.17i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(43.8717\)
Root analytic conductor: \(6.62357\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :6),\ 0.866 - 0.5i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.7696276289\)
\(L(\frac12)\) \(\approx\) \(0.7696276289\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 420. iT \)
good5 \( 1 + 2.12e4T + 2.44e8T^{2} \)
7 \( 1 - 7.11e3iT - 1.38e10T^{2} \)
11 \( 1 + 1.88e6iT - 3.13e12T^{2} \)
13 \( 1 + 8.56e6T + 2.32e13T^{2} \)
17 \( 1 - 6.77e6T + 5.82e14T^{2} \)
19 \( 1 - 6.45e5iT - 2.21e15T^{2} \)
23 \( 1 - 2.15e8iT - 2.19e16T^{2} \)
29 \( 1 + 3.66e8T + 3.53e17T^{2} \)
31 \( 1 + 1.03e9iT - 7.87e17T^{2} \)
37 \( 1 - 3.52e9T + 6.58e18T^{2} \)
41 \( 1 - 5.66e9T + 2.25e19T^{2} \)
43 \( 1 + 4.86e9iT - 3.99e19T^{2} \)
47 \( 1 - 1.89e10iT - 1.16e20T^{2} \)
53 \( 1 - 1.61e9T + 4.91e20T^{2} \)
59 \( 1 + 1.28e10iT - 1.77e21T^{2} \)
61 \( 1 - 3.86e10T + 2.65e21T^{2} \)
67 \( 1 - 1.56e11iT - 8.18e21T^{2} \)
71 \( 1 - 4.16e10iT - 1.64e22T^{2} \)
73 \( 1 - 1.00e11T + 2.29e22T^{2} \)
79 \( 1 + 1.99e11iT - 5.90e22T^{2} \)
83 \( 1 - 2.30e11iT - 1.06e23T^{2} \)
89 \( 1 + 4.73e11T + 2.46e23T^{2} \)
97 \( 1 - 9.29e11T + 6.93e23T^{2} \)
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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

−12.97834126147342944234318730524, −11.89924477698390033273380698995, −11.20939628128588868711368311321, −9.454486979112651099200546931985, −7.937999728147007346316472955834, −7.35643010147094057975504996780, −5.63895316128618963457542009899, −4.02872488606677458696190180753, −2.65952765720757215377310393232, −0.74265847474827524918982195345, 0.32444443615239857451950420280, 2.54836108643428779291332230687, 4.08742554032965335842829706067, 4.94011357321322201048170286252, 7.03348738984483564798196120960, 7.984987763389579595616967227301, 9.468865226549661592922881035748, 10.57897679994073219253534957221, 11.90276613640515587264050847165, 12.57459056576727840352118519138

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