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Properties

Label 2-2e4-16.5-c3-0-2
Degree $2$
Conductor $16$
Sign $0.730 + 0.682i$
Analytic cond. $0.944030$
Root an. cond. $0.971612$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.81 − 0.253i)2-s + (5.49 − 5.49i)3-s + (7.87 + 1.42i)4-s + (−4.66 − 4.66i)5-s + (−16.8 + 14.0i)6-s + 24.8i·7-s + (−21.8 − 6.00i)8-s − 33.4i·9-s + (11.9 + 14.3i)10-s + (22.3 + 22.3i)11-s + (51.1 − 35.4i)12-s + (−11.2 + 11.2i)13-s + (6.30 − 70.1i)14-s − 51.2·15-s + (59.9 + 22.4i)16-s − 88.4·17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0894i)2-s + (1.05 − 1.05i)3-s + (0.983 + 0.178i)4-s + (−0.417 − 0.417i)5-s + (−1.14 + 0.958i)6-s + 1.34i·7-s + (−0.964 − 0.265i)8-s − 1.23i·9-s + (0.378 + 0.452i)10-s + (0.612 + 0.612i)11-s + (1.22 − 0.852i)12-s + (−0.240 + 0.240i)13-s + (0.120 − 1.33i)14-s − 0.882·15-s + (0.936 + 0.350i)16-s − 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.730 + 0.682i$
Analytic conductor: \(0.944030\)
Root analytic conductor: \(0.971612\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :3/2),\ 0.730 + 0.682i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.799302 - 0.315141i\)
\(L(\frac12)\) \(\approx\) \(0.799302 - 0.315141i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.81 + 0.253i)T \)
good3 \( 1 + (-5.49 + 5.49i)T - 27iT^{2} \)
5 \( 1 + (4.66 + 4.66i)T + 125iT^{2} \)
7 \( 1 - 24.8iT - 343T^{2} \)
11 \( 1 + (-22.3 - 22.3i)T + 1.33e3iT^{2} \)
13 \( 1 + (11.2 - 11.2i)T - 2.19e3iT^{2} \)
17 \( 1 + 88.4T + 4.91e3T^{2} \)
19 \( 1 + (-37.8 + 37.8i)T - 6.85e3iT^{2} \)
23 \( 1 + 48.1iT - 1.21e4T^{2} \)
29 \( 1 + (-10.4 + 10.4i)T - 2.43e4iT^{2} \)
31 \( 1 + 96.9T + 2.97e4T^{2} \)
37 \( 1 + (163. + 163. i)T + 5.06e4iT^{2} \)
41 \( 1 - 360. iT - 6.89e4T^{2} \)
43 \( 1 + (100. + 100. i)T + 7.95e4iT^{2} \)
47 \( 1 - 220.T + 1.03e5T^{2} \)
53 \( 1 + (175. + 175. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-405. - 405. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-664. + 664. i)T - 2.26e5iT^{2} \)
67 \( 1 + (107. - 107. i)T - 3.00e5iT^{2} \)
71 \( 1 + 215. iT - 3.57e5T^{2} \)
73 \( 1 - 668. iT - 3.89e5T^{2} \)
79 \( 1 + 822.T + 4.93e5T^{2} \)
83 \( 1 + (-326. + 326. i)T - 5.71e5iT^{2} \)
89 \( 1 - 262. iT - 7.04e5T^{2} \)
97 \( 1 + 150.T + 9.12e5T^{2} \)
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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

−18.64532290781138716466997533540, −17.70774264941209122571400972689, −15.86252342172390217843419830064, −14.69988507753987762872501130772, −12.74861254480874746755156540661, −11.75882876176833103648751854754, −9.208322658304332383728951439167, −8.395618962321159118455322537640, −6.86678798802957610728685683430, −2.25265769513463768454853155610, 3.60641650265063205159219435507, 7.25148595476539696283598629308, 8.756413192025216190940882303818, 10.08574417320134893822762914855, 11.16800947255393671871768887243, 13.92801851059991218948762916483, 15.10379471695498216268016039356, 16.16604314025702061232247799118, 17.37080965920219412077854324230, 19.18378111044967489493928575706

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