↑

Properties

Label 2-48-16.11-c6-0-0
Degree $2$
Conductor $48$
Sign $-0.821 + 0.569i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−7.21 + 3.45i)2-s + (11.0 + 11.0i)3-s + (40.1 − 49.8i)4-s + (−26.8 − 26.8i)5-s + (−117. − 41.5i)6-s − 81.4·7-s + (−118. + 498. i)8-s + 242. i·9-s + (286. + 101. i)10-s + (−775. + 775. i)11-s + (991. − 106. i)12-s + (−35.8 + 35.8i)13-s + (587. − 281. i)14-s − 591. i·15-s + (−866. − 4.00e3i)16-s − 3.92e3·17-s + ⋯
L(s)  = 1  + (−0.902 + 0.431i)2-s + (0.408 + 0.408i)3-s + (0.627 − 0.778i)4-s + (−0.214 − 0.214i)5-s + (−0.544 − 0.192i)6-s − 0.237·7-s + (−0.230 + 0.973i)8-s + 0.333i·9-s + (0.286 + 0.101i)10-s + (−0.582 + 0.582i)11-s + (0.574 − 0.0614i)12-s + (−0.0163 + 0.0163i)13-s + (0.214 − 0.102i)14-s − 0.175i·15-s + (−0.211 − 0.977i)16-s − 0.799·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.821 + 0.569i$
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: \((2,\ 48,\ (\ :3),\ -0.821 + 0.569i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.00985006 - 0.0315089i\)
\(L(\frac12)\) \(\approx\) \(0.00985006 - 0.0315089i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.21 - 3.45i)T \)
3 \( 1 + (-11.0 - 11.0i)T \)
good5 \( 1 + (26.8 + 26.8i)T + 1.56e4iT^{2} \)
7 \( 1 + 81.4T + 1.17e5T^{2} \)
11 \( 1 + (775. - 775. i)T - 1.77e6iT^{2} \)
13 \( 1 + (35.8 - 35.8i)T - 4.82e6iT^{2} \)
17 \( 1 + 3.92e3T + 2.41e7T^{2} \)
19 \( 1 + (8.02e3 + 8.02e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 1.22e4T + 1.48e8T^{2} \)
29 \( 1 + (4.98e3 - 4.98e3i)T - 5.94e8iT^{2} \)
31 \( 1 - 1.91e3iT - 8.87e8T^{2} \)
37 \( 1 + (2.91e3 + 2.91e3i)T + 2.56e9iT^{2} \)
41 \( 1 + 4.73e3iT - 4.75e9T^{2} \)
43 \( 1 + (9.75e3 - 9.75e3i)T - 6.32e9iT^{2} \)
47 \( 1 - 1.90e5iT - 1.07e10T^{2} \)
53 \( 1 + (8.92e4 + 8.92e4i)T + 2.21e10iT^{2} \)
59 \( 1 + (6.55e4 - 6.55e4i)T - 4.21e10iT^{2} \)
61 \( 1 + (-4.68e4 + 4.68e4i)T - 5.15e10iT^{2} \)
67 \( 1 + (-1.56e5 - 1.56e5i)T + 9.04e10iT^{2} \)
71 \( 1 - 2.60e5T + 1.28e11T^{2} \)
73 \( 1 + 2.39e5iT - 1.51e11T^{2} \)
79 \( 1 - 8.45e5iT - 2.43e11T^{2} \)
83 \( 1 + (-1.24e5 - 1.24e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 9.21e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.39e6T + 8.32e11T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.45302406873356139083748396044, −14.29598145237118444435744117929, −12.81203385604813726652444351271, −11.17590941037723467767556465779, −10.09612463225593655463637622089, −8.978370901084254006164335338946, −7.918957151465482622344368303107, −6.49660255320784806810063083866, −4.66482278926816550295019652610, −2.32048713589467125458097306163, 0.01742426704034094039737443912, 2.03704479584654465318375805006, 3.59894954465072937258520515255, 6.35432784251513198548138666217, 7.75140953412833145623691445157, 8.690988429758043340887538830146, 10.08057338076375967508837479221, 11.19690704005620749871634118902, 12.44726074323194246747022964768, 13.46932601993762388158792406442

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