Properties

Degree $2$
Conductor $16$
Sign $0.794 + 0.607i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.96 + 0.523i)2-s + (5.54 − 5.54i)3-s + (15.4 − 4.15i)4-s + (21.7 − 21.7i)5-s + (−19.0 + 24.8i)6-s − 6.62·7-s + (−59.1 + 24.5i)8-s + 19.6i·9-s + (−74.8 + 97.5i)10-s + (−90.9 − 90.9i)11-s + (62.6 − 108. i)12-s + (221. + 221. i)13-s + (26.2 − 3.46i)14-s − 240. i·15-s + (221. − 128. i)16-s − 132.·17-s + ⋯
L(s)  = 1  + (−0.991 + 0.130i)2-s + (0.615 − 0.615i)3-s + (0.965 − 0.259i)4-s + (0.869 − 0.869i)5-s + (−0.529 + 0.690i)6-s − 0.135·7-s + (−0.923 + 0.383i)8-s + 0.242i·9-s + (−0.748 + 0.975i)10-s + (−0.752 − 0.752i)11-s + (0.434 − 0.754i)12-s + (1.31 + 1.31i)13-s + (0.133 − 0.0176i)14-s − 1.07i·15-s + (0.865 − 0.501i)16-s − 0.458·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.794 + 0.607i$
Motivic weight: \(4\)
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :2),\ 0.794 + 0.607i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.961139 - 0.325546i\)
\(L(\frac12)\) \(\approx\) \(0.961139 - 0.325546i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.96 - 0.523i)T \)
good3 \( 1 + (-5.54 + 5.54i)T - 81iT^{2} \)
5 \( 1 + (-21.7 + 21.7i)T - 625iT^{2} \)
7 \( 1 + 6.62T + 2.40e3T^{2} \)
11 \( 1 + (90.9 + 90.9i)T + 1.46e4iT^{2} \)
13 \( 1 + (-221. - 221. i)T + 2.85e4iT^{2} \)
17 \( 1 + 132.T + 8.35e4T^{2} \)
19 \( 1 + (402. - 402. i)T - 1.30e5iT^{2} \)
23 \( 1 - 27.5T + 2.79e5T^{2} \)
29 \( 1 + (-174. - 174. i)T + 7.07e5iT^{2} \)
31 \( 1 + 1.08e3iT - 9.23e5T^{2} \)
37 \( 1 + (-553. + 553. i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.80e3iT - 2.82e6T^{2} \)
43 \( 1 + (-17.8 - 17.8i)T + 3.41e6iT^{2} \)
47 \( 1 - 2.26e3iT - 4.87e6T^{2} \)
53 \( 1 + (822. - 822. i)T - 7.89e6iT^{2} \)
59 \( 1 + (972. + 972. i)T + 1.21e7iT^{2} \)
61 \( 1 + (2.05e3 + 2.05e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (-4.61e3 + 4.61e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 3.10e3T + 2.54e7T^{2} \)
73 \( 1 - 723. iT - 2.83e7T^{2} \)
79 \( 1 + 3.41e3iT - 3.89e7T^{2} \)
83 \( 1 + (161. - 161. i)T - 4.74e7iT^{2} \)
89 \( 1 + 1.46e3iT - 6.27e7T^{2} \)
97 \( 1 + 8.26e3T + 8.85e7T^{2} \)
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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

−18.48145365333977489144556971224, −16.93976221818018312331395328297, −16.06312600248498138586493785882, −14.03688302943436127153446761035, −12.92191042779052554890429860962, −10.89768072134304243385857239935, −9.131682571236201692823821504739, −8.168067195326538908324054644713, −6.14742840074135340869512901977, −1.81981118616267225115773413577, 2.83769239682419272724002276161, 6.50961575040248794836114343345, 8.517160603201234685407574685961, 9.952757274598534781543766354223, 10.79219537011454754166521516099, 13.08332737895783549310327026757, 14.95702252344881904016089672620, 15.71511193947199727622015293030, 17.68741720716460711210559180822, 18.18919493286855843706865738223

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