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Properties

Label 2-2e4-16.5-c5-0-6
Degree $2$
Conductor $16$
Sign $0.977 - 0.212i$
Analytic cond. $2.56614$
Root an. cond. $1.60191$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.55 + 3.34i)2-s + (14.5 − 14.5i)3-s + (9.55 + 30.5i)4-s + (−33.9 − 33.9i)5-s + (115. − 17.6i)6-s + 141. i·7-s + (−58.7 + 171. i)8-s − 181. i·9-s + (−41.0 − 268. i)10-s + (−509. − 509. i)11-s + (584. + 305. i)12-s + (0.484 − 0.484i)13-s + (−475. + 646. i)14-s − 989.·15-s + (−841. + 583. i)16-s + 1.55e3·17-s + ⋯
L(s)  = 1  + (0.805 + 0.592i)2-s + (0.935 − 0.935i)3-s + (0.298 + 0.954i)4-s + (−0.607 − 0.607i)5-s + (1.30 − 0.199i)6-s + 1.09i·7-s + (−0.324 + 0.945i)8-s − 0.748i·9-s + (−0.129 − 0.849i)10-s + (−1.27 − 1.27i)11-s + (1.17 + 0.613i)12-s + (0.000795 − 0.000795i)13-s + (−0.648 + 0.881i)14-s − 1.13·15-s + (−0.821 + 0.570i)16-s + 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.977 - 0.212i$
Analytic conductor: \(2.56614\)
Root analytic conductor: \(1.60191\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5/2),\ 0.977 - 0.212i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.19989 + 0.236271i\)
\(L(\frac12)\) \(\approx\) \(2.19989 + 0.236271i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.55 - 3.34i)T \)
good3 \( 1 + (-14.5 + 14.5i)T - 243iT^{2} \)
5 \( 1 + (33.9 + 33.9i)T + 3.12e3iT^{2} \)
7 \( 1 - 141. iT - 1.68e4T^{2} \)
11 \( 1 + (509. + 509. i)T + 1.61e5iT^{2} \)
13 \( 1 + (-0.484 + 0.484i)T - 3.71e5iT^{2} \)
17 \( 1 - 1.55e3T + 1.41e6T^{2} \)
19 \( 1 + (-92.6 + 92.6i)T - 2.47e6iT^{2} \)
23 \( 1 - 2.63e3iT - 6.43e6T^{2} \)
29 \( 1 + (-3.98e3 + 3.98e3i)T - 2.05e7iT^{2} \)
31 \( 1 + 3.88e3T + 2.86e7T^{2} \)
37 \( 1 + (-7.80e3 - 7.80e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 7.65e3iT - 1.15e8T^{2} \)
43 \( 1 + (3.94e3 + 3.94e3i)T + 1.47e8iT^{2} \)
47 \( 1 - 525.T + 2.29e8T^{2} \)
53 \( 1 + (1.55e4 + 1.55e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (-7.08e3 - 7.08e3i)T + 7.14e8iT^{2} \)
61 \( 1 + (1.93e4 - 1.93e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (2.11e4 - 2.11e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 2.40e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.10e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.33e4T + 3.07e9T^{2} \)
83 \( 1 + (4.95e4 - 4.95e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 6.72e4iT - 5.58e9T^{2} \)
97 \( 1 + 9.70e3T + 8.58e9T^{2} \)
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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

−18.33678370961223729985166470678, −16.35934240369600409170876759386, −15.35469435618899106854264233419, −13.93056958877731288126809431899, −12.91633397905077672058019907191, −11.83881517537217498920831315229, −8.504779822393819039289126338159, −7.80616545560816359744231217150, −5.58656866573475253349020079849, −2.93305839616342971971883819569, 3.12348431368310201547190038139, 4.52191157971060887941950053405, 7.46759724993281960547018867735, 9.916668746151687598138814257370, 10.70814281984752307393052916632, 12.66499969501720253390236373190, 14.26319118896204218297138112254, 14.97470776863036253904963147356, 16.12023348231626586039247066253, 18.46662185414462611803010212492

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