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Properties

Label 2-2e4-16.11-c10-0-10
Degree $2$
Conductor $16$
Sign $0.384 + 0.923i$
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  + (−6.25 − 31.3i)2-s + (196. + 196. i)3-s + (−945. + 392. i)4-s + (−1.81e3 − 1.81e3i)5-s + (4.94e3 − 7.40e3i)6-s + 2.23e4·7-s + (1.82e4 + 2.72e4i)8-s + 1.83e4i·9-s + (−4.55e4 + 6.82e4i)10-s + (2.11e5 − 2.11e5i)11-s + (−2.63e5 − 1.08e5i)12-s + (1.17e5 − 1.17e5i)13-s + (−1.39e5 − 7.01e5i)14-s − 7.13e5i·15-s + (7.40e5 − 7.42e5i)16-s − 1.27e6·17-s + ⋯
L(s)  = 1  + (−0.195 − 0.980i)2-s + (0.809 + 0.809i)3-s + (−0.923 + 0.383i)4-s + (−0.580 − 0.580i)5-s + (0.635 − 0.951i)6-s + 1.32·7-s + (0.556 + 0.830i)8-s + 0.310i·9-s + (−0.455 + 0.682i)10-s + (1.31 − 1.31i)11-s + (−1.05 − 0.437i)12-s + (0.316 − 0.316i)13-s + (−0.259 − 1.30i)14-s − 0.939i·15-s + (0.705 − 0.708i)16-s − 0.896·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.384 + 0.923i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ 0.384 + 0.923i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.65833 - 1.10597i\)
\(L(\frac12)\) \(\approx\) \(1.65833 - 1.10597i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.25 + 31.3i)T \)
good3 \( 1 + (-196. - 196. i)T + 5.90e4iT^{2} \)
5 \( 1 + (1.81e3 + 1.81e3i)T + 9.76e6iT^{2} \)
7 \( 1 - 2.23e4T + 2.82e8T^{2} \)
11 \( 1 + (-2.11e5 + 2.11e5i)T - 2.59e10iT^{2} \)
13 \( 1 + (-1.17e5 + 1.17e5i)T - 1.37e11iT^{2} \)
17 \( 1 + 1.27e6T + 2.01e12T^{2} \)
19 \( 1 + (-6.21e5 - 6.21e5i)T + 6.13e12iT^{2} \)
23 \( 1 - 6.08e6T + 4.14e13T^{2} \)
29 \( 1 + (8.09e6 - 8.09e6i)T - 4.20e14iT^{2} \)
31 \( 1 - 4.52e7iT - 8.19e14T^{2} \)
37 \( 1 + (-3.83e7 - 3.83e7i)T + 4.80e15iT^{2} \)
41 \( 1 + 1.45e8iT - 1.34e16T^{2} \)
43 \( 1 + (1.49e8 - 1.49e8i)T - 2.16e16iT^{2} \)
47 \( 1 + 1.28e8iT - 5.25e16T^{2} \)
53 \( 1 + (-7.42e7 - 7.42e7i)T + 1.74e17iT^{2} \)
59 \( 1 + (2.82e7 - 2.82e7i)T - 5.11e17iT^{2} \)
61 \( 1 + (4.27e8 - 4.27e8i)T - 7.13e17iT^{2} \)
67 \( 1 + (3.69e8 + 3.69e8i)T + 1.82e18iT^{2} \)
71 \( 1 + 3.22e9T + 3.25e18T^{2} \)
73 \( 1 + 1.76e9iT - 4.29e18T^{2} \)
79 \( 1 + 5.21e8iT - 9.46e18T^{2} \)
83 \( 1 + (-4.89e9 - 4.89e9i)T + 1.55e19iT^{2} \)
89 \( 1 - 1.22e9iT - 3.11e19T^{2} \)
97 \( 1 + 1.37e10T + 7.37e19T^{2} \)
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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

−16.49073619023199414918275189300, −14.82812092145502359613923430257, −13.79145586323001283418406121504, −11.90540803586178634853440076313, −10.86050784698870245650097433346, −8.941003938983340190279626061090, −8.426098686945147271018046327304, −4.63780615270523847353319598125, −3.43276561477777237051191731842, −1.14132610280561592230859175574, 1.63452249799961702826068460154, 4.40125221780341941338049358812, 6.91187858750043878560426447197, 7.77551457665421924055743736045, 9.110623960336754199537989031583, 11.39809380128099305688707813697, 13.30838580572172064083058560655, 14.61296094987619121833899067618, 15.06966652422059896585919623339, 17.12643995862755504757077064119

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