Properties

Degree $2$
Conductor $16$
Sign $-0.857 + 0.515i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.34 − 3.24i)2-s + (−4.63 − 4.63i)3-s + (−5.00 + 15.1i)4-s + (−29.2 − 29.2i)5-s + (−4.15 + 25.8i)6-s + 59.6·7-s + (60.9 − 19.3i)8-s − 38.0i·9-s + (−26.1 + 163. i)10-s + (−18.0 + 18.0i)11-s + (93.6 − 47.2i)12-s + (50.7 − 50.7i)13-s + (−139. − 193. i)14-s + 270. i·15-s + (−205. − 152. i)16-s − 223.·17-s + ⋯
L(s)  = 1  + (−0.586 − 0.810i)2-s + (−0.515 − 0.515i)3-s + (−0.313 + 0.949i)4-s + (−1.16 − 1.16i)5-s + (−0.115 + 0.719i)6-s + 1.21·7-s + (0.953 − 0.302i)8-s − 0.469i·9-s + (−0.261 + 1.63i)10-s + (−0.149 + 0.149i)11-s + (0.650 − 0.327i)12-s + (0.300 − 0.300i)13-s + (−0.713 − 0.985i)14-s + 1.20i·15-s + (−0.803 − 0.594i)16-s − 0.774·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.857 + 0.515i$
Motivic weight: \(4\)
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :2),\ -0.857 + 0.515i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.167025 - 0.602095i\)
\(L(\frac12)\) \(\approx\) \(0.167025 - 0.602095i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.34 + 3.24i)T \)
good3 \( 1 + (4.63 + 4.63i)T + 81iT^{2} \)
5 \( 1 + (29.2 + 29.2i)T + 625iT^{2} \)
7 \( 1 - 59.6T + 2.40e3T^{2} \)
11 \( 1 + (18.0 - 18.0i)T - 1.46e4iT^{2} \)
13 \( 1 + (-50.7 + 50.7i)T - 2.85e4iT^{2} \)
17 \( 1 + 223.T + 8.35e4T^{2} \)
19 \( 1 + (-14.7 - 14.7i)T + 1.30e5iT^{2} \)
23 \( 1 - 739.T + 2.79e5T^{2} \)
29 \( 1 + (-938. + 938. i)T - 7.07e5iT^{2} \)
31 \( 1 + 938. iT - 9.23e5T^{2} \)
37 \( 1 + (-263. - 263. i)T + 1.87e6iT^{2} \)
41 \( 1 - 248. iT - 2.82e6T^{2} \)
43 \( 1 + (-1.03e3 + 1.03e3i)T - 3.41e6iT^{2} \)
47 \( 1 - 2.01e3iT - 4.87e6T^{2} \)
53 \( 1 + (-833. - 833. i)T + 7.89e6iT^{2} \)
59 \( 1 + (2.22e3 - 2.22e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (341. - 341. i)T - 1.38e7iT^{2} \)
67 \( 1 + (-4.84e3 - 4.84e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 4.18e3T + 2.54e7T^{2} \)
73 \( 1 + 9.07e3iT - 2.83e7T^{2} \)
79 \( 1 - 735. iT - 3.89e7T^{2} \)
83 \( 1 + (-1.44e3 - 1.44e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 5.07e3iT - 6.27e7T^{2} \)
97 \( 1 + 2.52e3T + 8.85e7T^{2} \)
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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

−17.84424831119583119878803806045, −17.00205245477774661125487578705, −15.43997972945895984241752992716, −13.07426441537356061653501906523, −11.97597112646444880793895846938, −11.19421814224389576821305724521, −8.851294390512199938649562402821, −7.70603471298005074132102157777, −4.50242342143525858189527350076, −0.865822807810832524338782692320, 4.79570633276403909130541501185, 7.02113534111111739207514274888, 8.368308337485112717115875153783, 10.72986400653809419155181796425, 11.22564807435143651269661579608, 14.17082601783221074052722958633, 15.22052114135907929733526234869, 16.15229513641928954249784619848, 17.59314611754506436783854474420, 18.64234965369657600187396550767

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