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Properties

Label 2-2e4-16.3-c10-0-6
Degree $2$
Conductor $16$
Sign $0.835 - 0.549i$
Analytic cond. $10.1657$
Root an. cond. $3.18837$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−30.1 − 10.8i)2-s + (127. − 127. i)3-s + (789. + 652. i)4-s + (−1.08e3 + 1.08e3i)5-s + (−5.21e3 + 2.45e3i)6-s − 8.88e3·7-s + (−1.66e4 − 2.81e4i)8-s + 2.65e4i·9-s + (4.44e4 − 2.09e4i)10-s + (5.89e4 + 5.89e4i)11-s + (1.83e5 − 1.74e4i)12-s + (3.55e5 + 3.55e5i)13-s + (2.67e5 + 9.62e4i)14-s + 2.77e5i·15-s + (1.97e5 + 1.02e6i)16-s + 1.77e6·17-s + ⋯
L(s)  = 1  + (−0.940 − 0.338i)2-s + (0.524 − 0.524i)3-s + (0.770 + 0.637i)4-s + (−0.347 + 0.347i)5-s + (−0.671 + 0.315i)6-s − 0.528·7-s + (−0.509 − 0.860i)8-s + 0.449i·9-s + (0.444 − 0.209i)10-s + (0.365 + 0.365i)11-s + (0.738 − 0.0700i)12-s + (0.956 + 0.956i)13-s + (0.497 + 0.178i)14-s + 0.364i·15-s + (0.187 + 0.982i)16-s + 1.24·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.835 - 0.549i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ 0.835 - 0.549i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.10056 + 0.329515i\)
\(L(\frac12)\) \(\approx\) \(1.10056 + 0.329515i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (30.1 + 10.8i)T \)
good3 \( 1 + (-127. + 127. i)T - 5.90e4iT^{2} \)
5 \( 1 + (1.08e3 - 1.08e3i)T - 9.76e6iT^{2} \)
7 \( 1 + 8.88e3T + 2.82e8T^{2} \)
11 \( 1 + (-5.89e4 - 5.89e4i)T + 2.59e10iT^{2} \)
13 \( 1 + (-3.55e5 - 3.55e5i)T + 1.37e11iT^{2} \)
17 \( 1 - 1.77e6T + 2.01e12T^{2} \)
19 \( 1 + (-6.11e5 + 6.11e5i)T - 6.13e12iT^{2} \)
23 \( 1 + 2.27e6T + 4.14e13T^{2} \)
29 \( 1 + (-2.19e6 - 2.19e6i)T + 4.20e14iT^{2} \)
31 \( 1 - 6.85e6iT - 8.19e14T^{2} \)
37 \( 1 + (6.34e7 - 6.34e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 3.79e7iT - 1.34e16T^{2} \)
43 \( 1 + (-1.52e8 - 1.52e8i)T + 2.16e16iT^{2} \)
47 \( 1 + 1.83e8iT - 5.25e16T^{2} \)
53 \( 1 + (3.35e8 - 3.35e8i)T - 1.74e17iT^{2} \)
59 \( 1 + (6.55e8 + 6.55e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (8.33e8 + 8.33e8i)T + 7.13e17iT^{2} \)
67 \( 1 + (-1.04e9 + 1.04e9i)T - 1.82e18iT^{2} \)
71 \( 1 - 2.77e9T + 3.25e18T^{2} \)
73 \( 1 - 2.37e9iT - 4.29e18T^{2} \)
79 \( 1 - 5.94e9iT - 9.46e18T^{2} \)
83 \( 1 + (-2.01e9 + 2.01e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 1.11e9iT - 3.11e19T^{2} \)
97 \( 1 - 1.26e10T + 7.37e19T^{2} \)
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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

−16.90276068288983249335951211626, −15.76353399889662871508759202547, −14.00578904810558708129082050736, −12.46241874374891278076823881125, −11.06621848223748880448893250156, −9.479300477490329583637912599290, −8.023386768060421892454824552348, −6.76910240388076154522059847601, −3.31228739130563695414956096150, −1.53334487142813799818182757933, 0.75209683790875260558069614851, 3.38834296012395209914681191326, 6.02514576587346876723495208743, 8.003691060688062385368930654860, 9.184130656662387394841389073295, 10.41738610979320815571010847703, 12.18081440969247921545666581072, 14.27577475928495888226413469177, 15.59393407706837679375313899676, 16.34343312699154332239811960196

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