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Properties

Label 2-48-48.11-c3-0-11
Degree $2$
Conductor $48$
Sign $0.933 + 0.359i$
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  + (−1.97 + 2.01i)2-s + (−5.19 − 0.0749i)3-s + (−0.159 − 7.99i)4-s + (5.37 − 5.37i)5-s + (10.4 − 10.3i)6-s + 14.8·7-s + (16.4 + 15.5i)8-s + (26.9 + 0.779i)9-s + (0.214 + 21.5i)10-s + (−30.0 − 30.0i)11-s + (0.228 + 41.5i)12-s + (61.5 − 61.5i)13-s + (−29.4 + 30.0i)14-s + (−28.3 + 27.5i)15-s + (−63.9 + 2.55i)16-s − 48.8i·17-s + ⋯
L(s)  = 1  + (−0.700 + 0.714i)2-s + (−0.999 − 0.0144i)3-s + (−0.0199 − 0.999i)4-s + (0.480 − 0.480i)5-s + (0.710 − 0.703i)6-s + 0.802·7-s + (0.727 + 0.685i)8-s + (0.999 + 0.0288i)9-s + (0.00678 + 0.680i)10-s + (−0.823 − 0.823i)11-s + (0.00550 + 0.999i)12-s + (1.31 − 1.31i)13-s + (−0.561 + 0.572i)14-s + (−0.487 + 0.473i)15-s + (−0.999 + 0.0398i)16-s − 0.696i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.933 + 0.359i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ 0.933 + 0.359i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.772639 - 0.143491i\)
\(L(\frac12)\) \(\approx\) \(0.772639 - 0.143491i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.97 - 2.01i)T \)
3 \( 1 + (5.19 + 0.0749i)T \)
good5 \( 1 + (-5.37 + 5.37i)T - 125iT^{2} \)
7 \( 1 - 14.8T + 343T^{2} \)
11 \( 1 + (30.0 + 30.0i)T + 1.33e3iT^{2} \)
13 \( 1 + (-61.5 + 61.5i)T - 2.19e3iT^{2} \)
17 \( 1 + 48.8iT - 4.91e3T^{2} \)
19 \( 1 + (-7.45 - 7.45i)T + 6.85e3iT^{2} \)
23 \( 1 + 43.0iT - 1.21e4T^{2} \)
29 \( 1 + (-32.9 - 32.9i)T + 2.43e4iT^{2} \)
31 \( 1 - 173. iT - 2.97e4T^{2} \)
37 \( 1 + (177. + 177. i)T + 5.06e4iT^{2} \)
41 \( 1 - 454.T + 6.89e4T^{2} \)
43 \( 1 + (-239. + 239. i)T - 7.95e4iT^{2} \)
47 \( 1 + 30.4T + 1.03e5T^{2} \)
53 \( 1 + (235. - 235. i)T - 1.48e5iT^{2} \)
59 \( 1 + (260. + 260. i)T + 2.05e5iT^{2} \)
61 \( 1 + (388. - 388. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-334. - 334. i)T + 3.00e5iT^{2} \)
71 \( 1 - 522. iT - 3.57e5T^{2} \)
73 \( 1 - 689. iT - 3.89e5T^{2} \)
79 \( 1 + 692. iT - 4.93e5T^{2} \)
83 \( 1 + (-677. + 677. i)T - 5.71e5iT^{2} \)
89 \( 1 + 261.T + 7.04e5T^{2} \)
97 \( 1 + 641.T + 9.12e5T^{2} \)
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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

−15.67840639227266207400973516092, −14.02515448600761746711215674279, −12.88989738501175442045554860200, −11.10570145460803382509594857216, −10.48530314377798071959339507945, −8.850592151915488040487159211480, −7.64296884905255108893226858489, −5.90641283637423814951530333648, −5.15797704549718188308826143349, −0.967860463821695415958000930504, 1.76606116248357682761236837638, 4.40359651651583940657361752248, 6.38295236122156610514688643800, 7.85851236361295819000555853729, 9.538965841415425250545686040725, 10.71073210209915665411711204192, 11.36405400647796035893500662958, 12.59109466435361678280687418807, 13.80732750205645861947417799003, 15.56017605131350415916999491433

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