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Properties

Label 2-2e4-16.5-c5-0-3
Degree $2$
Conductor $16$
Sign $0.338 - 0.941i$
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  + (5.25 + 2.10i)2-s + (−12.7 + 12.7i)3-s + (23.1 + 22.0i)4-s + (39.8 + 39.8i)5-s + (−93.6 + 40.0i)6-s − 248. i·7-s + (75.2 + 164. i)8-s − 81.1i·9-s + (125. + 292. i)10-s + (−84.8 − 84.8i)11-s + (−575. + 13.7i)12-s + (453. − 453. i)13-s + (522. − 1.30e3i)14-s − 1.01e3·15-s + (48.7 + 1.02e3i)16-s − 336.·17-s + ⋯
L(s)  = 1  + (0.928 + 0.371i)2-s + (−0.816 + 0.816i)3-s + (0.723 + 0.690i)4-s + (0.712 + 0.712i)5-s + (−1.06 + 0.454i)6-s − 1.91i·7-s + (0.415 + 0.909i)8-s − 0.333i·9-s + (0.396 + 0.925i)10-s + (−0.211 − 0.211i)11-s + (−1.15 + 0.0275i)12-s + (0.744 − 0.744i)13-s + (0.712 − 1.77i)14-s − 1.16·15-s + (0.0476 + 0.998i)16-s − 0.282·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 - 0.941i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.338 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.338 - 0.941i$
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: \((2,\ 16,\ (\ :5/2),\ 0.338 - 0.941i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.52119 + 1.06973i\)
\(L(\frac12)\) \(\approx\) \(1.52119 + 1.06973i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.25 - 2.10i)T \)
good3 \( 1 + (12.7 - 12.7i)T - 243iT^{2} \)
5 \( 1 + (-39.8 - 39.8i)T + 3.12e3iT^{2} \)
7 \( 1 + 248. iT - 1.68e4T^{2} \)
11 \( 1 + (84.8 + 84.8i)T + 1.61e5iT^{2} \)
13 \( 1 + (-453. + 453. i)T - 3.71e5iT^{2} \)
17 \( 1 + 336.T + 1.41e6T^{2} \)
19 \( 1 + (-25.1 + 25.1i)T - 2.47e6iT^{2} \)
23 \( 1 - 1.83e3iT - 6.43e6T^{2} \)
29 \( 1 + (2.14e3 - 2.14e3i)T - 2.05e7iT^{2} \)
31 \( 1 + 6.17e3T + 2.86e7T^{2} \)
37 \( 1 + (2.56e3 + 2.56e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.78e4iT - 1.15e8T^{2} \)
43 \( 1 + (-6.72e3 - 6.72e3i)T + 1.47e8iT^{2} \)
47 \( 1 - 8.63e3T + 2.29e8T^{2} \)
53 \( 1 + (-2.10e4 - 2.10e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (-6.06e3 - 6.06e3i)T + 7.14e8iT^{2} \)
61 \( 1 + (229. - 229. i)T - 8.44e8iT^{2} \)
67 \( 1 + (1.14e4 - 1.14e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 7.92e3iT - 1.80e9T^{2} \)
73 \( 1 - 5.22e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.61e4T + 3.07e9T^{2} \)
83 \( 1 + (3.25e4 - 3.25e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 6.90e4iT - 5.58e9T^{2} \)
97 \( 1 - 6.24e4T + 8.58e9T^{2} \)
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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

−17.70094512797033933980887028662, −16.82773079426351095994334646534, −15.77170803143487004775897902085, −14.17029051702304553731950680213, −13.26766390662132771244248894534, −10.97694906882933506991663003476, −10.46837816647335606996466406880, −7.26764908984990395815725109193, −5.67058769811675056050423558521, −3.88549879968674950636724850218, 1.89250025774099711101763045315, 5.36255597225008349884410989271, 6.31391872269958727951777391265, 9.173140717294694950301655076679, 11.43578909493681190667828642451, 12.39187377210766067270833234946, 13.22847790024342925940206942542, 15.00940875197052418807870394487, 16.41050229440714813807902544006, 18.09384707678809353489709345252

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