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Properties

Label 2-2e4-16.3-c8-0-4
Degree $2$
Conductor $16$
Sign $0.603 - 0.797i$
Analytic cond. $6.51805$
Root an. cond. $2.55304$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.02 − 13.8i)2-s + (17.4 − 17.4i)3-s + (−127. + 222. i)4-s + (−396. + 396. i)5-s + (−381. − 101. i)6-s + 432.·7-s + (4.09e3 − 18.8i)8-s + 5.95e3i·9-s + (8.66e3 + 2.30e3i)10-s + (8.01e3 + 8.01e3i)11-s + (1.65e3 + 6.09e3i)12-s + (1.27e4 + 1.27e4i)13-s + (−3.47e3 − 5.99e3i)14-s + 1.38e4i·15-s + (−3.31e4 − 5.65e4i)16-s − 8.10e4·17-s + ⋯
L(s)  = 1  + (−0.501 − 0.865i)2-s + (0.215 − 0.215i)3-s + (−0.497 + 0.867i)4-s + (−0.634 + 0.634i)5-s + (−0.294 − 0.0783i)6-s + 0.180·7-s + (0.999 − 0.00459i)8-s + 0.907i·9-s + (0.866 + 0.230i)10-s + (0.547 + 0.547i)11-s + (0.0796 + 0.293i)12-s + (0.445 + 0.445i)13-s + (−0.0903 − 0.155i)14-s + 0.272i·15-s + (−0.505 − 0.862i)16-s − 0.970·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.603 - 0.797i$
Analytic conductor: \(6.51805\)
Root analytic conductor: \(2.55304\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :4),\ 0.603 - 0.797i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.796591 + 0.395873i\)
\(L(\frac12)\) \(\approx\) \(0.796591 + 0.395873i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (8.02 + 13.8i)T \)
good3 \( 1 + (-17.4 + 17.4i)T - 6.56e3iT^{2} \)
5 \( 1 + (396. - 396. i)T - 3.90e5iT^{2} \)
7 \( 1 - 432.T + 5.76e6T^{2} \)
11 \( 1 + (-8.01e3 - 8.01e3i)T + 2.14e8iT^{2} \)
13 \( 1 + (-1.27e4 - 1.27e4i)T + 8.15e8iT^{2} \)
17 \( 1 + 8.10e4T + 6.97e9T^{2} \)
19 \( 1 + (1.29e5 - 1.29e5i)T - 1.69e10iT^{2} \)
23 \( 1 - 1.31e3T + 7.83e10T^{2} \)
29 \( 1 + (4.47e5 + 4.47e5i)T + 5.00e11iT^{2} \)
31 \( 1 - 1.21e6iT - 8.52e11T^{2} \)
37 \( 1 + (-1.65e6 + 1.65e6i)T - 3.51e12iT^{2} \)
41 \( 1 + 3.85e6iT - 7.98e12T^{2} \)
43 \( 1 + (-2.19e6 - 2.19e6i)T + 1.16e13iT^{2} \)
47 \( 1 + 3.58e6iT - 2.38e13T^{2} \)
53 \( 1 + (-6.45e6 + 6.45e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (-1.37e7 - 1.37e7i)T + 1.46e14iT^{2} \)
61 \( 1 + (-7.47e6 - 7.47e6i)T + 1.91e14iT^{2} \)
67 \( 1 + (9.70e6 - 9.70e6i)T - 4.06e14iT^{2} \)
71 \( 1 + 2.18e7T + 6.45e14T^{2} \)
73 \( 1 - 4.21e7iT - 8.06e14T^{2} \)
79 \( 1 + 7.15e7iT - 1.51e15T^{2} \)
83 \( 1 + (-4.46e7 + 4.46e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 6.27e7iT - 3.93e15T^{2} \)
97 \( 1 + 3.58e7T + 7.83e15T^{2} \)
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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

−17.74114334400256660583723238419, −16.32672748963027174736716512004, −14.54581674743037305128925572660, −13.08597076537371212911733950456, −11.58369630481829656574195956573, −10.52429988635946856770553472963, −8.684999477463888428271954245569, −7.27705476483760200271155323904, −4.05550989591237625800102892364, −2.02378946858815191622895256372, 0.58972348121070319535939322535, 4.32332400631313513760624599526, 6.39386492565139559502983478332, 8.283244162116644060355997072321, 9.255863639731739753769389565382, 11.21008011728117900068088368654, 13.13635575061518075861771241679, 14.80477978530797736668707770843, 15.68614932688700149817962449262, 16.92112130614533304489286684394

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