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Properties

Label 2-48-16.3-c6-0-1
Degree $2$
Conductor $48$
Sign $-0.792 + 0.609i$
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  + (4.87 + 6.34i)2-s + (−11.0 + 11.0i)3-s + (−16.5 + 61.8i)4-s + (−95.7 + 95.7i)5-s + (−123. − 16.2i)6-s + 338.·7-s + (−472. + 196. i)8-s − 242. i·9-s + (−1.07e3 − 141. i)10-s + (−1.59e3 − 1.59e3i)11-s + (−499. − 863. i)12-s + (602. + 602. i)13-s + (1.64e3 + 2.14e3i)14-s − 2.11e3i·15-s + (−3.54e3 − 2.04e3i)16-s − 1.41e3·17-s + ⋯
L(s)  = 1  + (0.608 + 0.793i)2-s + (−0.408 + 0.408i)3-s + (−0.258 + 0.966i)4-s + (−0.766 + 0.766i)5-s + (−0.572 − 0.0752i)6-s + 0.987·7-s + (−0.923 + 0.383i)8-s − 0.333i·9-s + (−1.07 − 0.141i)10-s + (−1.19 − 1.19i)11-s + (−0.288 − 0.499i)12-s + (0.274 + 0.274i)13-s + (0.601 + 0.783i)14-s − 0.625i·15-s + (−0.866 − 0.499i)16-s − 0.287·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.792 + 0.609i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ -0.792 + 0.609i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.324615 - 0.955066i\)
\(L(\frac12)\) \(\approx\) \(0.324615 - 0.955066i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.87 - 6.34i)T \)
3 \( 1 + (11.0 - 11.0i)T \)
good5 \( 1 + (95.7 - 95.7i)T - 1.56e4iT^{2} \)
7 \( 1 - 338.T + 1.17e5T^{2} \)
11 \( 1 + (1.59e3 + 1.59e3i)T + 1.77e6iT^{2} \)
13 \( 1 + (-602. - 602. i)T + 4.82e6iT^{2} \)
17 \( 1 + 1.41e3T + 2.41e7T^{2} \)
19 \( 1 + (6.64e3 - 6.64e3i)T - 4.70e7iT^{2} \)
23 \( 1 - 1.49e4T + 1.48e8T^{2} \)
29 \( 1 + (2.56e4 + 2.56e4i)T + 5.94e8iT^{2} \)
31 \( 1 - 5.22e4iT - 8.87e8T^{2} \)
37 \( 1 + (1.12e4 - 1.12e4i)T - 2.56e9iT^{2} \)
41 \( 1 - 7.80e4iT - 4.75e9T^{2} \)
43 \( 1 + (-6.29e4 - 6.29e4i)T + 6.32e9iT^{2} \)
47 \( 1 - 5.91e4iT - 1.07e10T^{2} \)
53 \( 1 + (4.56e4 - 4.56e4i)T - 2.21e10iT^{2} \)
59 \( 1 + (-4.01e4 - 4.01e4i)T + 4.21e10iT^{2} \)
61 \( 1 + (6.80e4 + 6.80e4i)T + 5.15e10iT^{2} \)
67 \( 1 + (-7.58e4 + 7.58e4i)T - 9.04e10iT^{2} \)
71 \( 1 + 5.20e5T + 1.28e11T^{2} \)
73 \( 1 + 3.49e5iT - 1.51e11T^{2} \)
79 \( 1 - 5.84e5iT - 2.43e11T^{2} \)
83 \( 1 + (7.55e4 - 7.55e4i)T - 3.26e11iT^{2} \)
89 \( 1 + 8.81e5iT - 4.96e11T^{2} \)
97 \( 1 - 4.63e5T + 8.32e11T^{2} \)
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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

−15.12137608474986644468856566434, −14.28173338639474789775044880012, −12.95625674237485451467287091465, −11.44010880743218675672558989743, −10.83450147500761210359741855567, −8.561826356140874013384737216826, −7.58296954301895537101525557465, −6.06228559510966399053069394969, −4.71276235433789096947973952802, −3.25753043864545520400747932257, 0.38407656725825352067974627011, 2.09355761130387371589492477140, 4.42275866412068593524908279524, 5.27957228427353231734814538810, 7.37200503907839101180648025166, 8.814567972487372213056883425512, 10.62436800118288754314136075786, 11.45108073391439696145936854620, 12.64248705304929988175759952370, 13.16978530192788299429406110311

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