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Properties

Label 2-2e4-16.13-c9-0-3
Degree $2$
Conductor $16$
Sign $0.252 - 0.967i$
Analytic cond. $8.24057$
Root an. cond. $2.87063$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.9 + 19.2i)2-s + (−128. − 128. i)3-s + (−227. − 458. i)4-s + (−28.4 + 28.4i)5-s + (4.00e3 − 940. i)6-s + 2.52e3i·7-s + (1.15e4 + 1.07e3i)8-s + 1.33e4i·9-s + (−208. − 886. i)10-s + (−3.56e4 + 3.56e4i)11-s + (−2.96e4 + 8.82e4i)12-s + (6.87e4 + 6.87e4i)13-s + (−4.85e4 − 3.01e4i)14-s + 7.31e3·15-s + (−1.58e5 + 2.09e5i)16-s + 6.74e5·17-s + ⋯
L(s)  = 1  + (−0.526 + 0.850i)2-s + (−0.916 − 0.916i)3-s + (−0.445 − 0.895i)4-s + (−0.0203 + 0.0203i)5-s + (1.26 − 0.296i)6-s + 0.397i·7-s + (0.995 + 0.0930i)8-s + 0.679i·9-s + (−0.00658 − 0.0280i)10-s + (−0.733 + 0.733i)11-s + (−0.412 + 1.22i)12-s + (0.667 + 0.667i)13-s + (−0.338 − 0.209i)14-s + 0.0373·15-s + (−0.603 + 0.797i)16-s + 1.95·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(8.24057\)
Root analytic conductor: \(2.87063\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :9/2),\ 0.252 - 0.967i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.587344 + 0.453789i\)
\(L(\frac12)\) \(\approx\) \(0.587344 + 0.453789i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (11.9 - 19.2i)T \)
good3 \( 1 + (128. + 128. i)T + 1.96e4iT^{2} \)
5 \( 1 + (28.4 - 28.4i)T - 1.95e6iT^{2} \)
7 \( 1 - 2.52e3iT - 4.03e7T^{2} \)
11 \( 1 + (3.56e4 - 3.56e4i)T - 2.35e9iT^{2} \)
13 \( 1 + (-6.87e4 - 6.87e4i)T + 1.06e10iT^{2} \)
17 \( 1 - 6.74e5T + 1.18e11T^{2} \)
19 \( 1 + (-9.03e4 - 9.03e4i)T + 3.22e11iT^{2} \)
23 \( 1 + 1.00e6iT - 1.80e12T^{2} \)
29 \( 1 + (9.25e5 + 9.25e5i)T + 1.45e13iT^{2} \)
31 \( 1 - 7.04e6T + 2.64e13T^{2} \)
37 \( 1 + (6.03e6 - 6.03e6i)T - 1.29e14iT^{2} \)
41 \( 1 - 2.11e7iT - 3.27e14T^{2} \)
43 \( 1 + (1.29e7 - 1.29e7i)T - 5.02e14iT^{2} \)
47 \( 1 + 5.61e7T + 1.11e15T^{2} \)
53 \( 1 + (2.19e5 - 2.19e5i)T - 3.29e15iT^{2} \)
59 \( 1 + (-7.91e7 + 7.91e7i)T - 8.66e15iT^{2} \)
61 \( 1 + (-1.01e8 - 1.01e8i)T + 1.16e16iT^{2} \)
67 \( 1 + (-9.87e6 - 9.87e6i)T + 2.72e16iT^{2} \)
71 \( 1 - 3.99e8iT - 4.58e16T^{2} \)
73 \( 1 - 2.61e7iT - 5.88e16T^{2} \)
79 \( 1 - 3.28e8T + 1.19e17T^{2} \)
83 \( 1 + (9.37e7 + 9.37e7i)T + 1.86e17iT^{2} \)
89 \( 1 + 4.96e8iT - 3.50e17T^{2} \)
97 \( 1 - 9.71e8T + 7.60e17T^{2} \)
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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

−17.28237874789617819570248767590, −16.22016606454547721774264000843, −14.73825793988314592793439562264, −13.11436239337243413189686317474, −11.69821091848215279641239378510, −9.947216461132219337188525827497, −8.001234379420106375287323949872, −6.64379544067124369490254485200, −5.35039219792418219454845169757, −1.24353915029210042882705609746, 0.60923821694946134933422437351, 3.50262181979697394176246013172, 5.35579697703842226460671209544, 8.065642353744574736178068277441, 10.01392985833185109312959831694, 10.75109053640148174853091085955, 12.02207171483150296470730374506, 13.65491353311118982132052430909, 15.89329496610710803767651358286, 16.74758687624506725477992269270

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