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Properties

Label 2-2e4-16.5-c7-0-5
Degree $2$
Conductor $16$
Sign $0.471 - 0.881i$
Analytic cond. $4.99816$
Root an. cond. $2.23565$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10.5 − 4.07i)2-s + (−49.4 + 49.4i)3-s + (94.8 − 85.9i)4-s + (252. + 252. i)5-s + (−320. + 723. i)6-s + 1.48e3i·7-s + (651. − 1.29e3i)8-s − 2.70e3i·9-s + (3.68e3 + 1.63e3i)10-s + (−2.89e3 − 2.89e3i)11-s + (−440. + 8.94e3i)12-s + (3.03e3 − 3.03e3i)13-s + (6.03e3 + 1.56e4i)14-s − 2.49e4·15-s + (1.60e3 − 1.63e4i)16-s + 6.39e3·17-s + ⋯
L(s)  = 1  + (0.933 − 0.359i)2-s + (−1.05 + 1.05i)3-s + (0.741 − 0.671i)4-s + (0.901 + 0.901i)5-s + (−0.606 + 1.36i)6-s + 1.63i·7-s + (0.449 − 0.893i)8-s − 1.23i·9-s + (1.16 + 0.516i)10-s + (−0.655 − 0.655i)11-s + (−0.0735 + 1.49i)12-s + (0.382 − 0.382i)13-s + (0.587 + 1.52i)14-s − 1.90·15-s + (0.0982 − 0.995i)16-s + 0.315·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(4.99816\)
Root analytic conductor: \(2.23565\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :7/2),\ 0.471 - 0.881i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.79093 + 1.07313i\)
\(L(\frac12)\) \(\approx\) \(1.79093 + 1.07313i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-10.5 + 4.07i)T \)
good3 \( 1 + (49.4 - 49.4i)T - 2.18e3iT^{2} \)
5 \( 1 + (-252. - 252. i)T + 7.81e4iT^{2} \)
7 \( 1 - 1.48e3iT - 8.23e5T^{2} \)
11 \( 1 + (2.89e3 + 2.89e3i)T + 1.94e7iT^{2} \)
13 \( 1 + (-3.03e3 + 3.03e3i)T - 6.27e7iT^{2} \)
17 \( 1 - 6.39e3T + 4.10e8T^{2} \)
19 \( 1 + (-2.13e4 + 2.13e4i)T - 8.93e8iT^{2} \)
23 \( 1 - 7.31e4iT - 3.40e9T^{2} \)
29 \( 1 + (-8.44e4 + 8.44e4i)T - 1.72e10iT^{2} \)
31 \( 1 - 1.04e5T + 2.75e10T^{2} \)
37 \( 1 + (-4.70e4 - 4.70e4i)T + 9.49e10iT^{2} \)
41 \( 1 + 3.28e5iT - 1.94e11T^{2} \)
43 \( 1 + (4.61e4 + 4.61e4i)T + 2.71e11iT^{2} \)
47 \( 1 - 4.95e5T + 5.06e11T^{2} \)
53 \( 1 + (1.02e5 + 1.02e5i)T + 1.17e12iT^{2} \)
59 \( 1 + (1.53e6 + 1.53e6i)T + 2.48e12iT^{2} \)
61 \( 1 + (1.20e6 - 1.20e6i)T - 3.14e12iT^{2} \)
67 \( 1 + (1.41e6 - 1.41e6i)T - 6.06e12iT^{2} \)
71 \( 1 + 1.12e6iT - 9.09e12T^{2} \)
73 \( 1 - 5.53e6iT - 1.10e13T^{2} \)
79 \( 1 + 3.15e6T + 1.92e13T^{2} \)
83 \( 1 + (-6.06e6 + 6.06e6i)T - 2.71e13iT^{2} \)
89 \( 1 - 1.19e7iT - 4.42e13T^{2} \)
97 \( 1 + 8.32e6T + 8.07e13T^{2} \)
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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

−17.74784384600196782752053240963, −15.87292243311867498260904730667, −15.28142663101920526915113778766, −13.67803077242493986353049905546, −11.89821512571572444920642987880, −10.87557513591227129625949591558, −9.726466198215211776775979514810, −6.01757369986670920872591199071, −5.35884301538957262023118009783, −2.85150178249173382191651533968, 1.29467805700263515577759874195, 4.80590264269299462821236903959, 6.29825975487870518465145359142, 7.58515220636476780020590404754, 10.54210896925606387790848984030, 12.22796792031131529542354595968, 13.14993745268305571769929430897, 13.99211624459442258015677339049, 16.43554775251438449673889897902, 17.01189914049121502832741382024

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