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Properties

Label 2-2e4-16.5-c9-0-5
Degree $2$
Conductor $16$
Sign $0.970 + 0.242i$
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  + (−15.3 + 16.5i)2-s + (−149. + 149. i)3-s + (−37.7 − 510. i)4-s + (−1.56e3 − 1.56e3i)5-s + (−176. − 4.77e3i)6-s + 1.07e4i·7-s + (9.04e3 + 7.23e3i)8-s − 2.49e4i·9-s + (4.99e4 − 1.84e3i)10-s + (−6.14e3 − 6.14e3i)11-s + (8.19e4 + 7.06e4i)12-s + (2.47e4 − 2.47e4i)13-s + (−1.78e5 − 1.65e5i)14-s + 4.66e5·15-s + (−2.59e5 + 3.86e4i)16-s + 1.20e5·17-s + ⋯
L(s)  = 1  + (−0.680 + 0.732i)2-s + (−1.06 + 1.06i)3-s + (−0.0738 − 0.997i)4-s + (−1.11 − 1.11i)5-s + (−0.0556 − 1.50i)6-s + 1.69i·7-s + (0.780 + 0.624i)8-s − 1.26i·9-s + (1.57 − 0.0583i)10-s + (−0.126 − 0.126i)11-s + (1.14 + 0.983i)12-s + (0.240 − 0.240i)13-s + (−1.24 − 1.15i)14-s + 2.37·15-s + (−0.989 + 0.147i)16-s + 0.349·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(8.24057\)
Root analytic conductor: \(2.87063\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :9/2),\ 0.970 + 0.242i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.331342 - 0.0407792i\)
\(L(\frac12)\) \(\approx\) \(0.331342 - 0.0407792i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (15.3 - 16.5i)T \)
good3 \( 1 + (149. - 149. i)T - 1.96e4iT^{2} \)
5 \( 1 + (1.56e3 + 1.56e3i)T + 1.95e6iT^{2} \)
7 \( 1 - 1.07e4iT - 4.03e7T^{2} \)
11 \( 1 + (6.14e3 + 6.14e3i)T + 2.35e9iT^{2} \)
13 \( 1 + (-2.47e4 + 2.47e4i)T - 1.06e10iT^{2} \)
17 \( 1 - 1.20e5T + 1.18e11T^{2} \)
19 \( 1 + (-2.62e5 + 2.62e5i)T - 3.22e11iT^{2} \)
23 \( 1 + 1.04e6iT - 1.80e12T^{2} \)
29 \( 1 + (-3.76e6 + 3.76e6i)T - 1.45e13iT^{2} \)
31 \( 1 + 7.20e6T + 2.64e13T^{2} \)
37 \( 1 + (1.64e6 + 1.64e6i)T + 1.29e14iT^{2} \)
41 \( 1 - 1.08e7iT - 3.27e14T^{2} \)
43 \( 1 + (-1.01e7 - 1.01e7i)T + 5.02e14iT^{2} \)
47 \( 1 - 3.94e7T + 1.11e15T^{2} \)
53 \( 1 + (4.93e7 + 4.93e7i)T + 3.29e15iT^{2} \)
59 \( 1 + (1.29e7 + 1.29e7i)T + 8.66e15iT^{2} \)
61 \( 1 + (-6.60e7 + 6.60e7i)T - 1.16e16iT^{2} \)
67 \( 1 + (6.78e7 - 6.78e7i)T - 2.72e16iT^{2} \)
71 \( 1 + 2.87e8iT - 4.58e16T^{2} \)
73 \( 1 + 6.48e7iT - 5.88e16T^{2} \)
79 \( 1 - 2.38e8T + 1.19e17T^{2} \)
83 \( 1 + (2.05e7 - 2.05e7i)T - 1.86e17iT^{2} \)
89 \( 1 + 1.42e8iT - 3.50e17T^{2} \)
97 \( 1 + 1.06e9T + 7.60e17T^{2} \)
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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

−16.48196182088779924591714888642, −15.88045474645932773946095846535, −15.14692688397208918223291351258, −12.28013996120966905218385887521, −11.18196394118908986983365681444, −9.413087980847677028679672973093, −8.298374308255534328147773199659, −5.76895065420129289468836674354, −4.75553272387044841490465807728, −0.31346050822095609244870287289, 1.05682518442749894464936801925, 3.69265500390260267933696300040, 7.05165857648062304177310840255, 7.55610058807879570947948724009, 10.51627341830330394371607686458, 11.23662284347669424165871259618, 12.37537137594433262116266798747, 13.88460413680776318935319011506, 16.24020973240054624556646579527, 17.37768894887208433999497673211

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