Properties

Label 2-48-16.11-c6-0-15
Degree $2$
Conductor $48$
Sign $0.478 - 0.878i$
Analytic cond. $11.0425$
Root an. cond. $3.32304$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.80 + 1.76i)2-s + (11.0 + 11.0i)3-s + (57.7 + 27.5i)4-s + (38.1 + 38.1i)5-s + (66.5 + 105. i)6-s + 34.3·7-s + (401. + 317. i)8-s + 242. i·9-s + (229. + 364. i)10-s + (−45.9 + 45.9i)11-s + (332. + 940. i)12-s + (−248. + 248. i)13-s + (267. + 60.6i)14-s + 840. i·15-s + (2.57e3 + 3.18e3i)16-s + 552.·17-s + ⋯
L(s)  = 1  + (0.975 + 0.221i)2-s + (0.408 + 0.408i)3-s + (0.902 + 0.431i)4-s + (0.304 + 0.304i)5-s + (0.307 + 0.488i)6-s + 0.100·7-s + (0.784 + 0.620i)8-s + 0.333i·9-s + (0.229 + 0.364i)10-s + (−0.0344 + 0.0344i)11-s + (0.192 + 0.544i)12-s + (−0.112 + 0.112i)13-s + (0.0975 + 0.0221i)14-s + 0.248i·15-s + (0.628 + 0.778i)16-s + 0.112·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.878i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.478 - 0.878i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ 0.478 - 0.878i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.23248 + 1.91961i\)
\(L(\frac12)\) \(\approx\) \(3.23248 + 1.91961i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.80 - 1.76i)T \)
3 \( 1 + (-11.0 - 11.0i)T \)
good5 \( 1 + (-38.1 - 38.1i)T + 1.56e4iT^{2} \)
7 \( 1 - 34.3T + 1.17e5T^{2} \)
11 \( 1 + (45.9 - 45.9i)T - 1.77e6iT^{2} \)
13 \( 1 + (248. - 248. i)T - 4.82e6iT^{2} \)
17 \( 1 - 552.T + 2.41e7T^{2} \)
19 \( 1 + (-606. - 606. i)T + 4.70e7iT^{2} \)
23 \( 1 - 562.T + 1.48e8T^{2} \)
29 \( 1 + (-3.37e3 + 3.37e3i)T - 5.94e8iT^{2} \)
31 \( 1 + 1.84e4iT - 8.87e8T^{2} \)
37 \( 1 + (2.46e4 + 2.46e4i)T + 2.56e9iT^{2} \)
41 \( 1 + 7.36e4iT - 4.75e9T^{2} \)
43 \( 1 + (-2.12e4 + 2.12e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 1.59e5iT - 1.07e10T^{2} \)
53 \( 1 + (1.48e5 + 1.48e5i)T + 2.21e10iT^{2} \)
59 \( 1 + (-1.88e5 + 1.88e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (2.17e5 - 2.17e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (-3.15e5 - 3.15e5i)T + 9.04e10iT^{2} \)
71 \( 1 - 2.59e5T + 1.28e11T^{2} \)
73 \( 1 - 6.16e5iT - 1.51e11T^{2} \)
79 \( 1 - 5.07e5iT - 2.43e11T^{2} \)
83 \( 1 + (-6.13e5 - 6.13e5i)T + 3.26e11iT^{2} \)
89 \( 1 - 3.35e5iT - 4.96e11T^{2} \)
97 \( 1 + 9.27e5T + 8.32e11T^{2} \)
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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

−14.48552425658407049297191284052, −13.72776384979968717067615986499, −12.48485963098932567861603972862, −11.19080230263703834051533891131, −9.972164043130692123676325204388, −8.261629722906943731711426337554, −6.83153140835274555084984729913, −5.34917080681522804576560651834, −3.85998195761022316763563566931, −2.31810202795931140966301722062, 1.47512446544284778045052845531, 3.11926857055993954397098450575, 4.88303175260577176552201422062, 6.33901859319801456728067210272, 7.75302921106775653595921850869, 9.449008778738583906841015694418, 10.91136479783609634375418245641, 12.20214306330479641829676141787, 13.11828175261866888089872753789, 14.04547784851931295508524891477

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