Properties

Degree $2$
Conductor $16$
Sign $0.969 - 0.245i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.89 + 0.921i)2-s + (−0.0461 − 0.0461i)3-s + (14.3 + 7.17i)4-s + (−8.04 − 8.04i)5-s + (−0.137 − 0.222i)6-s − 49.8·7-s + (49.0 + 41.0i)8-s − 80.9i·9-s + (−23.8 − 38.7i)10-s + (−84.2 + 84.2i)11-s + (−0.329 − 0.992i)12-s + (19.4 − 19.4i)13-s + (−194. − 45.9i)14-s + 0.743i·15-s + (153. + 205. i)16-s + 437.·17-s + ⋯
L(s)  = 1  + (0.973 + 0.230i)2-s + (−0.00513 − 0.00513i)3-s + (0.893 + 0.448i)4-s + (−0.321 − 0.321i)5-s + (−0.00381 − 0.00617i)6-s − 1.01·7-s + (0.766 + 0.642i)8-s − 0.999i·9-s + (−0.238 − 0.387i)10-s + (−0.696 + 0.696i)11-s + (−0.00228 − 0.00688i)12-s + (0.115 − 0.115i)13-s + (−0.990 − 0.234i)14-s + 0.00330i·15-s + (0.598 + 0.801i)16-s + 1.51·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.969 - 0.245i$
Motivic weight: \(4\)
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :2),\ 0.969 - 0.245i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.74823 + 0.218174i\)
\(L(\frac12)\) \(\approx\) \(1.74823 + 0.218174i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.89 - 0.921i)T \)
good3 \( 1 + (0.0461 + 0.0461i)T + 81iT^{2} \)
5 \( 1 + (8.04 + 8.04i)T + 625iT^{2} \)
7 \( 1 + 49.8T + 2.40e3T^{2} \)
11 \( 1 + (84.2 - 84.2i)T - 1.46e4iT^{2} \)
13 \( 1 + (-19.4 + 19.4i)T - 2.85e4iT^{2} \)
17 \( 1 - 437.T + 8.35e4T^{2} \)
19 \( 1 + (-349. - 349. i)T + 1.30e5iT^{2} \)
23 \( 1 + 404.T + 2.79e5T^{2} \)
29 \( 1 + (1.03e3 - 1.03e3i)T - 7.07e5iT^{2} \)
31 \( 1 + 1.50e3iT - 9.23e5T^{2} \)
37 \( 1 + (434. + 434. i)T + 1.87e6iT^{2} \)
41 \( 1 - 696. iT - 2.82e6T^{2} \)
43 \( 1 + (-917. + 917. i)T - 3.41e6iT^{2} \)
47 \( 1 - 111. iT - 4.87e6T^{2} \)
53 \( 1 + (-1.04e3 - 1.04e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (1.71e3 - 1.71e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (-3.71e3 + 3.71e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (1.85e3 + 1.85e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 1.16e3T + 2.54e7T^{2} \)
73 \( 1 + 905. iT - 2.83e7T^{2} \)
79 \( 1 - 5.86e3iT - 3.89e7T^{2} \)
83 \( 1 + (7.56e3 + 7.56e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 6.43e3iT - 6.27e7T^{2} \)
97 \( 1 + 413.T + 8.85e7T^{2} \)
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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

−18.44214232242615362776676512345, −16.63865646857689478148577717734, −15.70637470858463905991663312817, −14.44153424946255885505268784229, −12.79790951785769041336484127087, −12.05463413331696985869012825249, −9.914494113394808781524437225248, −7.58251392551289159851261695615, −5.81383502305343207188944363452, −3.57984947593880779818833706187, 3.19579339065940467130538594777, 5.54809941886057873762844597646, 7.45422617270577692587380342603, 10.13013851536242749291181585369, 11.47262576612651886195658694260, 13.02463675152856957467615779582, 14.02167179307889253899707651541, 15.68918650799043334934786482130, 16.45383940866912285793185124088, 18.85402178372611661923925899795

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