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Properties

Label 2-2e4-16.3-c6-0-7
Degree $2$
Conductor $16$
Sign $0.975 - 0.220i$
Analytic cond. $3.68086$
Root an. cond. $1.91855$
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  + (7.25 + 3.37i)2-s + (14.8 − 14.8i)3-s + (41.2 + 48.9i)4-s + (41.0 − 41.0i)5-s + (158. − 57.7i)6-s − 67.2·7-s + (133. + 494. i)8-s + 285. i·9-s + (435. − 159. i)10-s + (−959. − 959. i)11-s + (1.34e3 + 115. i)12-s + (−1.15e3 − 1.15e3i)13-s + (−487. − 226. i)14-s − 1.22e3i·15-s + (−696. + 4.03e3i)16-s − 8.74e3·17-s + ⋯
L(s)  = 1  + (0.906 + 0.421i)2-s + (0.551 − 0.551i)3-s + (0.644 + 0.764i)4-s + (0.328 − 0.328i)5-s + (0.732 − 0.267i)6-s − 0.196·7-s + (0.261 + 0.965i)8-s + 0.391i·9-s + (0.435 − 0.159i)10-s + (−0.720 − 0.720i)11-s + (0.777 + 0.0665i)12-s + (−0.524 − 0.524i)13-s + (−0.177 − 0.0827i)14-s − 0.362i·15-s + (−0.169 + 0.985i)16-s − 1.78·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.975 - 0.220i$
Analytic conductor: \(3.68086\)
Root analytic conductor: \(1.91855\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :3),\ 0.975 - 0.220i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.59444 + 0.289005i\)
\(L(\frac12)\) \(\approx\) \(2.59444 + 0.289005i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.25 - 3.37i)T \)
good3 \( 1 + (-14.8 + 14.8i)T - 729iT^{2} \)
5 \( 1 + (-41.0 + 41.0i)T - 1.56e4iT^{2} \)
7 \( 1 + 67.2T + 1.17e5T^{2} \)
11 \( 1 + (959. + 959. i)T + 1.77e6iT^{2} \)
13 \( 1 + (1.15e3 + 1.15e3i)T + 4.82e6iT^{2} \)
17 \( 1 + 8.74e3T + 2.41e7T^{2} \)
19 \( 1 + (-2.78e3 + 2.78e3i)T - 4.70e7iT^{2} \)
23 \( 1 - 1.80e4T + 1.48e8T^{2} \)
29 \( 1 + (-1.49e4 - 1.49e4i)T + 5.94e8iT^{2} \)
31 \( 1 + 3.31e4iT - 8.87e8T^{2} \)
37 \( 1 + (4.36e3 - 4.36e3i)T - 2.56e9iT^{2} \)
41 \( 1 + 2.35e4iT - 4.75e9T^{2} \)
43 \( 1 + (-6.16e4 - 6.16e4i)T + 6.32e9iT^{2} \)
47 \( 1 + 1.35e5iT - 1.07e10T^{2} \)
53 \( 1 + (-1.73e5 + 1.73e5i)T - 2.21e10iT^{2} \)
59 \( 1 + (4.91e4 + 4.91e4i)T + 4.21e10iT^{2} \)
61 \( 1 + (-1.90e5 - 1.90e5i)T + 5.15e10iT^{2} \)
67 \( 1 + (2.46e5 - 2.46e5i)T - 9.04e10iT^{2} \)
71 \( 1 + 5.72e3T + 1.28e11T^{2} \)
73 \( 1 - 3.44e5iT - 1.51e11T^{2} \)
79 \( 1 - 7.85e4iT - 2.43e11T^{2} \)
83 \( 1 + (-4.72e5 + 4.72e5i)T - 3.26e11iT^{2} \)
89 \( 1 + 4.18e5iT - 4.96e11T^{2} \)
97 \( 1 + 7.58e5T + 8.32e11T^{2} \)
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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

−17.64214258361500712653603491637, −16.27734066736252497737106511946, −15.00352801814481136728722638522, −13.39697825708219416443469857381, −13.08479693603125211564324792459, −11.08365464521690195687279468932, −8.598775435386475422089012529790, −7.10913357216116933293892147752, −5.17114502792778189696594804040, −2.65062918024158642333562218180, 2.64601104547139974863311474580, 4.56020044533902974775716321148, 6.71993881450919304869570856618, 9.387601398460391390117834471618, 10.65490174670902324785894097916, 12.36945401281111959257007079956, 13.76367784467897541111718653002, 14.93473941151050788312807140948, 15.82102094913555157545593261065, 17.86788926103715178953618445683

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