Properties

Degree $2$
Conductor $16$
Sign $0.895 - 0.445i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.59 + 3.66i)2-s + (11.5 − 11.5i)3-s + (−10.9 + 11.6i)4-s + (−14.6 + 14.6i)5-s + (60.6 + 23.8i)6-s − 24.0·7-s + (−60.3 − 21.3i)8-s − 184. i·9-s + (−76.8 − 30.2i)10-s + (61.7 + 61.7i)11-s + (8.99 + 260. i)12-s + (−37.5 − 37.5i)13-s + (−38.2 − 88.1i)14-s + 336. i·15-s + (−17.6 − 255. i)16-s + 96.8·17-s + ⋯
L(s)  = 1  + (0.398 + 0.917i)2-s + (1.28 − 1.28i)3-s + (−0.682 + 0.731i)4-s + (−0.584 + 0.584i)5-s + (1.68 + 0.663i)6-s − 0.490·7-s + (−0.942 − 0.334i)8-s − 2.27i·9-s + (−0.768 − 0.302i)10-s + (0.510 + 0.510i)11-s + (0.0624 + 1.80i)12-s + (−0.222 − 0.222i)13-s + (−0.195 − 0.449i)14-s + 1.49i·15-s + (−0.0689 − 0.997i)16-s + 0.335·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.895 - 0.445i$
Motivic weight: \(4\)
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :2),\ 0.895 - 0.445i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.60166 + 0.376500i\)
\(L(\frac12)\) \(\approx\) \(1.60166 + 0.376500i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.59 - 3.66i)T \)
good3 \( 1 + (-11.5 + 11.5i)T - 81iT^{2} \)
5 \( 1 + (14.6 - 14.6i)T - 625iT^{2} \)
7 \( 1 + 24.0T + 2.40e3T^{2} \)
11 \( 1 + (-61.7 - 61.7i)T + 1.46e4iT^{2} \)
13 \( 1 + (37.5 + 37.5i)T + 2.85e4iT^{2} \)
17 \( 1 - 96.8T + 8.35e4T^{2} \)
19 \( 1 + (156. - 156. i)T - 1.30e5iT^{2} \)
23 \( 1 - 959.T + 2.79e5T^{2} \)
29 \( 1 + (350. + 350. i)T + 7.07e5iT^{2} \)
31 \( 1 - 237. iT - 9.23e5T^{2} \)
37 \( 1 + (560. - 560. i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.80e3iT - 2.82e6T^{2} \)
43 \( 1 + (-206. - 206. i)T + 3.41e6iT^{2} \)
47 \( 1 + 1.59e3iT - 4.87e6T^{2} \)
53 \( 1 + (2.23e3 - 2.23e3i)T - 7.89e6iT^{2} \)
59 \( 1 + (-2.35e3 - 2.35e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (4.44e3 + 4.44e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (3.99e3 - 3.99e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 4.92e3T + 2.54e7T^{2} \)
73 \( 1 + 2.65e3iT - 2.83e7T^{2} \)
79 \( 1 + 8.79e3iT - 3.89e7T^{2} \)
83 \( 1 + (228. - 228. i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.05e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.10e4T + 8.85e7T^{2} \)
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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

−18.67362143382087843869620817233, −17.23197139507562589712522624123, −15.25237178346426632208636044805, −14.55080387615144459262081701660, −13.27082143254654524369379200458, −12.25629850445580637020621486542, −9.092123188883225691197934584075, −7.64628811669246102234889741563, −6.73955090828850978221781892691, −3.31899641249790776604097509629, 3.24136632779191059008346759910, 4.63321888821984903121493299752, 8.641861570286521097512944579674, 9.570241631106630927038022129565, 11.07623545870791620437147909776, 12.91680420581687463946079406359, 14.27464466343625529987408896839, 15.30834055529718772232564129899, 16.58551666090320423692259634048, 19.17584595974085617836731236789

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