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Properties

Label 2-48-48.35-c3-0-0
Degree $2$
Conductor $48$
Sign $-0.950 + 0.311i$
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  + (−0.775 + 2.72i)2-s + (−2.65 + 4.46i)3-s + (−6.79 − 4.21i)4-s + (5.27 + 5.27i)5-s + (−10.0 − 10.6i)6-s − 22.9·7-s + (16.7 − 15.2i)8-s + (−12.9 − 23.7i)9-s + (−18.4 + 10.2i)10-s + (−10.6 + 10.6i)11-s + (36.8 − 19.1i)12-s + (12.7 + 12.7i)13-s + (17.7 − 62.3i)14-s + (−37.5 + 9.57i)15-s + (28.3 + 57.3i)16-s + 134. i·17-s + ⋯
L(s)  = 1  + (−0.274 + 0.961i)2-s + (−0.510 + 0.859i)3-s + (−0.849 − 0.527i)4-s + (0.472 + 0.472i)5-s + (−0.686 − 0.726i)6-s − 1.23·7-s + (0.740 − 0.672i)8-s + (−0.478 − 0.878i)9-s + (−0.583 + 0.324i)10-s + (−0.291 + 0.291i)11-s + (0.887 − 0.461i)12-s + (0.271 + 0.271i)13-s + (0.339 − 1.19i)14-s + (−0.646 + 0.164i)15-s + (0.443 + 0.896i)16-s + 1.91i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.950 + 0.311i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ -0.950 + 0.311i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0979102 - 0.613534i\)
\(L(\frac12)\) \(\approx\) \(0.0979102 - 0.613534i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.775 - 2.72i)T \)
3 \( 1 + (2.65 - 4.46i)T \)
good5 \( 1 + (-5.27 - 5.27i)T + 125iT^{2} \)
7 \( 1 + 22.9T + 343T^{2} \)
11 \( 1 + (10.6 - 10.6i)T - 1.33e3iT^{2} \)
13 \( 1 + (-12.7 - 12.7i)T + 2.19e3iT^{2} \)
17 \( 1 - 134. iT - 4.91e3T^{2} \)
19 \( 1 + (46.9 - 46.9i)T - 6.85e3iT^{2} \)
23 \( 1 - 93.7iT - 1.21e4T^{2} \)
29 \( 1 + (-161. + 161. i)T - 2.43e4iT^{2} \)
31 \( 1 - 120. iT - 2.97e4T^{2} \)
37 \( 1 + (-2.42 + 2.42i)T - 5.06e4iT^{2} \)
41 \( 1 + 253.T + 6.89e4T^{2} \)
43 \( 1 + (135. + 135. i)T + 7.95e4iT^{2} \)
47 \( 1 - 468.T + 1.03e5T^{2} \)
53 \( 1 + (-321. - 321. i)T + 1.48e5iT^{2} \)
59 \( 1 + (119. - 119. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-310. - 310. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-705. + 705. i)T - 3.00e5iT^{2} \)
71 \( 1 - 501. iT - 3.57e5T^{2} \)
73 \( 1 - 641. iT - 3.89e5T^{2} \)
79 \( 1 + 1.23e3iT - 4.93e5T^{2} \)
83 \( 1 + (-87.3 - 87.3i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.19e3T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−15.76536732161635804637725163228, −15.07087032983912924587893029700, −13.79898961875820076631080696810, −12.49876040885783449622756001925, −10.43175082850671817634607391534, −9.967650886786838334908882033807, −8.591919300156645414945730915509, −6.58906506809900313939343815046, −5.83824717278603508413803815597, −3.94317512520753635886314456149, 0.55528101737796334534504951778, 2.74534372722940205296616776696, 5.19721080892675146525149712002, 6.88911590145714419397310607935, 8.609553923766304521266714284964, 9.811894413511394838516607080079, 11.09777648560103638234770795217, 12.31775689519782787474564263102, 13.13506150214681103403288224668, 13.80470889140523670144211500742

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