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Properties

Label 2-2e4-16.3-c12-0-20
Degree $2$
Conductor $16$
Sign $0.437 + 0.899i$
Analytic cond. $14.6239$
Root an. cond. $3.82412$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (63.9 − 0.953i)2-s + (386. − 386. i)3-s + (4.09e3 − 122. i)4-s + (2.08e4 − 2.08e4i)5-s + (2.43e4 − 2.50e4i)6-s − 1.71e5·7-s + (2.61e5 − 1.17e4i)8-s + 2.32e5i·9-s + (1.31e6 − 1.35e6i)10-s + (−1.44e6 − 1.44e6i)11-s + (1.53e6 − 1.62e6i)12-s + (2.99e6 + 2.99e6i)13-s + (−1.10e7 + 1.64e5i)14-s − 1.60e7i·15-s + (1.67e7 − 9.99e5i)16-s + 1.21e7·17-s + ⋯
L(s)  = 1  + (0.999 − 0.0149i)2-s + (0.529 − 0.529i)3-s + (0.999 − 0.0298i)4-s + (1.33 − 1.33i)5-s + (0.521 − 0.537i)6-s − 1.46·7-s + (0.999 − 0.0446i)8-s + 0.438i·9-s + (1.31 − 1.35i)10-s + (−0.813 − 0.813i)11-s + (0.513 − 0.545i)12-s + (0.621 + 0.621i)13-s + (−1.46 + 0.0217i)14-s − 1.41i·15-s + (0.998 − 0.0595i)16-s + 0.504·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.437 + 0.899i$
Analytic conductor: \(14.6239\)
Root analytic conductor: \(3.82412\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :6),\ 0.437 + 0.899i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(3.67704 - 2.30143i\)
\(L(\frac12)\) \(\approx\) \(3.67704 - 2.30143i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-63.9 + 0.953i)T \)
good3 \( 1 + (-386. + 386. i)T - 5.31e5iT^{2} \)
5 \( 1 + (-2.08e4 + 2.08e4i)T - 2.44e8iT^{2} \)
7 \( 1 + 1.71e5T + 1.38e10T^{2} \)
11 \( 1 + (1.44e6 + 1.44e6i)T + 3.13e12iT^{2} \)
13 \( 1 + (-2.99e6 - 2.99e6i)T + 2.32e13iT^{2} \)
17 \( 1 - 1.21e7T + 5.82e14T^{2} \)
19 \( 1 + (7.67e6 - 7.67e6i)T - 2.21e15iT^{2} \)
23 \( 1 - 6.46e7T + 2.19e16T^{2} \)
29 \( 1 + (-4.40e8 - 4.40e8i)T + 3.53e17iT^{2} \)
31 \( 1 - 3.88e8iT - 7.87e17T^{2} \)
37 \( 1 + (2.64e9 - 2.64e9i)T - 6.58e18iT^{2} \)
41 \( 1 - 5.60e9iT - 2.25e19T^{2} \)
43 \( 1 + (3.47e9 + 3.47e9i)T + 3.99e19iT^{2} \)
47 \( 1 + 4.68e9iT - 1.16e20T^{2} \)
53 \( 1 + (4.63e9 - 4.63e9i)T - 4.91e20iT^{2} \)
59 \( 1 + (-3.98e10 - 3.98e10i)T + 1.77e21iT^{2} \)
61 \( 1 + (4.20e10 + 4.20e10i)T + 2.65e21iT^{2} \)
67 \( 1 + (-2.77e10 + 2.77e10i)T - 8.18e21iT^{2} \)
71 \( 1 + 7.43e10T + 1.64e22T^{2} \)
73 \( 1 - 2.20e10iT - 2.29e22T^{2} \)
79 \( 1 + 1.33e11iT - 5.90e22T^{2} \)
83 \( 1 + (2.04e11 - 2.04e11i)T - 1.06e23iT^{2} \)
89 \( 1 + 3.01e11iT - 2.46e23T^{2} \)
97 \( 1 - 2.75e11T + 6.93e23T^{2} \)
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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

−16.11747120203548011446524225197, −13.83166246892623586716313620263, −13.34827941727664933062377570927, −12.53299701127114628510233076737, −10.22439247978232373523093013273, −8.588811867442683501791164855912, −6.45585426837910474822920175301, −5.21072436849995511082774480212, −2.94507980667674148048733573031, −1.41835958829917301118958554324, 2.54149031107246090342316457070, 3.41420386509571797300753336454, 5.83186873506605152449381138232, 6.90265476616225767953715510344, 9.766478108812437076114551521604, 10.49755085250972445427639780064, 12.75268556518931193911189714064, 13.76443581749531283308108397976, 14.94360756315289260345796416746, 15.79857358463275783740403747371

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