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Properties

Label 2-2e4-16.3-c8-0-0
Degree $2$
Conductor $16$
Sign $-0.0274 + 0.999i$
Analytic cond. $6.51805$
Root an. cond. $2.55304$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−15.9 − 1.67i)2-s + (−93.3 + 93.3i)3-s + (250. + 53.3i)4-s + (−695. + 695. i)5-s + (1.64e3 − 1.32e3i)6-s + 805.·7-s + (−3.89e3 − 1.26e3i)8-s − 1.08e4i·9-s + (1.22e4 − 9.89e3i)10-s + (−1.19e4 − 1.19e4i)11-s + (−2.83e4 + 1.83e4i)12-s + (2.45e4 + 2.45e4i)13-s + (−1.28e4 − 1.35e3i)14-s − 1.29e5i·15-s + (5.98e4 + 2.67e4i)16-s + 8.61e4·17-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (−1.15 + 1.15i)3-s + (0.978 + 0.208i)4-s + (−1.11 + 1.11i)5-s + (1.26 − 1.02i)6-s + 0.335·7-s + (−0.950 − 0.309i)8-s − 1.65i·9-s + (1.22 − 0.989i)10-s + (−0.817 − 0.817i)11-s + (−1.36 + 0.887i)12-s + (0.861 + 0.861i)13-s + (−0.333 − 0.0351i)14-s − 2.56i·15-s + (0.913 + 0.407i)16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0274 + 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0274 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.0274 + 0.999i$
Analytic conductor: \(6.51805\)
Root analytic conductor: \(2.55304\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :4),\ -0.0274 + 0.999i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.00371759 - 0.00382122i\)
\(L(\frac12)\) \(\approx\) \(0.00371759 - 0.00382122i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (15.9 + 1.67i)T \)
good3 \( 1 + (93.3 - 93.3i)T - 6.56e3iT^{2} \)
5 \( 1 + (695. - 695. i)T - 3.90e5iT^{2} \)
7 \( 1 - 805.T + 5.76e6T^{2} \)
11 \( 1 + (1.19e4 + 1.19e4i)T + 2.14e8iT^{2} \)
13 \( 1 + (-2.45e4 - 2.45e4i)T + 8.15e8iT^{2} \)
17 \( 1 - 8.61e4T + 6.97e9T^{2} \)
19 \( 1 + (6.42e4 - 6.42e4i)T - 1.69e10iT^{2} \)
23 \( 1 + 1.93e5T + 7.83e10T^{2} \)
29 \( 1 + (3.52e5 + 3.52e5i)T + 5.00e11iT^{2} \)
31 \( 1 + 9.10e5iT - 8.52e11T^{2} \)
37 \( 1 + (5.12e4 - 5.12e4i)T - 3.51e12iT^{2} \)
41 \( 1 - 4.16e6iT - 7.98e12T^{2} \)
43 \( 1 + (1.53e6 + 1.53e6i)T + 1.16e13iT^{2} \)
47 \( 1 - 1.86e5iT - 2.38e13T^{2} \)
53 \( 1 + (-8.08e6 + 8.08e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (1.04e7 + 1.04e7i)T + 1.46e14iT^{2} \)
61 \( 1 + (-2.57e6 - 2.57e6i)T + 1.91e14iT^{2} \)
67 \( 1 + (-7.07e6 + 7.07e6i)T - 4.06e14iT^{2} \)
71 \( 1 + 1.58e7T + 6.45e14T^{2} \)
73 \( 1 - 4.16e6iT - 8.06e14T^{2} \)
79 \( 1 + 5.49e7iT - 1.51e15T^{2} \)
83 \( 1 + (2.38e7 - 2.38e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 2.35e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.07e8T + 7.83e15T^{2} \)
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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

−18.28050693361324558702331115542, −16.66729319587409656620972395167, −15.96125747204176108325459272835, −14.89946248952367090727523591030, −11.65560378038933904228625395302, −11.12162941363498226214051014713, −10.09283744169955285911465813042, −8.011125760811945290692849009005, −6.14743848855659397268490065651, −3.66078319341679115371052936813, 0.00533596407071888730027963976, 1.24475541790312902368950344304, 5.44778191978982871525112141478, 7.36587338111530083886396598772, 8.309046477810237314049551985767, 10.72193506729124553554673914736, 12.00393020193686414985656745666, 12.74280414759160725136871639525, 15.53558658159476936873557705243, 16.55618650347257246890160330285

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