Properties

Degree $2$
Conductor $16$
Sign $0.0625 - 0.998i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.329 + 3.98i)2-s + (3.91 + 3.91i)3-s + (−15.7 + 2.62i)4-s + (4.72 + 4.72i)5-s + (−14.3 + 16.8i)6-s + 45.3·7-s + (−15.6 − 62.0i)8-s − 50.3i·9-s + (−17.2 + 20.3i)10-s + (110. − 110. i)11-s + (−72.0 − 51.4i)12-s + (−157. + 157. i)13-s + (14.9 + 180. i)14-s + 36.9i·15-s + (242. − 82.9i)16-s − 378.·17-s + ⋯
L(s)  = 1  + (0.0824 + 0.996i)2-s + (0.434 + 0.434i)3-s + (−0.986 + 0.164i)4-s + (0.188 + 0.188i)5-s + (−0.397 + 0.469i)6-s + 0.925·7-s + (−0.245 − 0.969i)8-s − 0.621i·9-s + (−0.172 + 0.203i)10-s + (0.910 − 0.910i)11-s + (−0.500 − 0.357i)12-s + (−0.929 + 0.929i)13-s + (0.0763 + 0.922i)14-s + 0.164i·15-s + (0.946 − 0.324i)16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0625 - 0.998i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0625 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.0625 - 0.998i$
Motivic weight: \(4\)
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :2),\ 0.0625 - 0.998i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.972083 + 0.913051i\)
\(L(\frac12)\) \(\approx\) \(0.972083 + 0.913051i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.329 - 3.98i)T \)
good3 \( 1 + (-3.91 - 3.91i)T + 81iT^{2} \)
5 \( 1 + (-4.72 - 4.72i)T + 625iT^{2} \)
7 \( 1 - 45.3T + 2.40e3T^{2} \)
11 \( 1 + (-110. + 110. i)T - 1.46e4iT^{2} \)
13 \( 1 + (157. - 157. i)T - 2.85e4iT^{2} \)
17 \( 1 + 378.T + 8.35e4T^{2} \)
19 \( 1 + (-203. - 203. i)T + 1.30e5iT^{2} \)
23 \( 1 + 740.T + 2.79e5T^{2} \)
29 \( 1 + (-82.6 + 82.6i)T - 7.07e5iT^{2} \)
31 \( 1 - 286. iT - 9.23e5T^{2} \)
37 \( 1 + (-1.47e3 - 1.47e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 1.30e3iT - 2.82e6T^{2} \)
43 \( 1 + (366. - 366. i)T - 3.41e6iT^{2} \)
47 \( 1 - 751. iT - 4.87e6T^{2} \)
53 \( 1 + (1.92e3 + 1.92e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (1.35e3 - 1.35e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (1.83e3 - 1.83e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (-2.20e3 - 2.20e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 8.97e3T + 2.54e7T^{2} \)
73 \( 1 - 9.35e3iT - 2.83e7T^{2} \)
79 \( 1 + 2.86e3iT - 3.89e7T^{2} \)
83 \( 1 + (1.03e3 + 1.03e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 5.17e3iT - 6.27e7T^{2} \)
97 \( 1 - 8.53e3T + 8.85e7T^{2} \)
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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

−18.32017907925411956637064304594, −17.26027638646566460055406876955, −15.94414423095316621165681243396, −14.54251279602776663467650347052, −14.03091836890127686429486001996, −11.84374927557888680869242683254, −9.614757238684959758651764610605, −8.373225946522111000334548853210, −6.44594916667510073489493164342, −4.27606256340579136856730734517, 2.02900317240427470257613452903, 4.78698080693059237819645710626, 7.84733366107618129527469502856, 9.467487768139053480495048633211, 11.12973582163581992529551236727, 12.53153195734836419831062782250, 13.76637874289020613983941106080, 14.90588827770738879389788562891, 17.33300259254849257495955728118, 18.13029345470954437944677333996

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