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Properties

Label 2-2e4-16.5-c11-0-17
Degree $2$
Conductor $16$
Sign $0.212 + 0.977i$
Analytic cond. $12.2934$
Root an. cond. $3.50620$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (43.9 + 10.8i)2-s + (221. − 221. i)3-s + (1.81e3 + 949. i)4-s + (−7.91e3 − 7.91e3i)5-s + (1.21e4 − 7.34e3i)6-s − 7.68e4i·7-s + (6.94e4 + 6.13e4i)8-s + 7.90e4i·9-s + (−2.62e5 − 4.33e5i)10-s + (−1.97e5 − 1.97e5i)11-s + (6.12e5 − 1.91e5i)12-s + (−4.68e4 + 4.68e4i)13-s + (8.30e5 − 3.37e6i)14-s − 3.50e6·15-s + (2.39e6 + 3.44e6i)16-s + 9.74e6·17-s + ⋯
L(s)  = 1  + (0.971 + 0.238i)2-s + (0.526 − 0.526i)3-s + (0.885 + 0.463i)4-s + (−1.13 − 1.13i)5-s + (0.636 − 0.385i)6-s − 1.72i·7-s + (0.749 + 0.661i)8-s + 0.446i·9-s + (−0.829 − 1.37i)10-s + (−0.370 − 0.370i)11-s + (0.710 − 0.222i)12-s + (−0.0350 + 0.0350i)13-s + (0.412 − 1.67i)14-s − 1.19·15-s + (0.569 + 0.821i)16-s + 1.66·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.212 + 0.977i$
Analytic conductor: \(12.2934\)
Root analytic conductor: \(3.50620\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :11/2),\ 0.212 + 0.977i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.41824 - 1.94957i\)
\(L(\frac12)\) \(\approx\) \(2.41824 - 1.94957i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-43.9 - 10.8i)T \)
good3 \( 1 + (-221. + 221. i)T - 1.77e5iT^{2} \)
5 \( 1 + (7.91e3 + 7.91e3i)T + 4.88e7iT^{2} \)
7 \( 1 + 7.68e4iT - 1.97e9T^{2} \)
11 \( 1 + (1.97e5 + 1.97e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (4.68e4 - 4.68e4i)T - 1.79e12iT^{2} \)
17 \( 1 - 9.74e6T + 3.42e13T^{2} \)
19 \( 1 + (-2.85e6 + 2.85e6i)T - 1.16e14iT^{2} \)
23 \( 1 + 3.12e7iT - 9.52e14T^{2} \)
29 \( 1 + (1.26e8 - 1.26e8i)T - 1.22e16iT^{2} \)
31 \( 1 - 3.46e7T + 2.54e16T^{2} \)
37 \( 1 + (-2.25e8 - 2.25e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 4.55e8iT - 5.50e17T^{2} \)
43 \( 1 + (-7.03e8 - 7.03e8i)T + 9.29e17iT^{2} \)
47 \( 1 - 6.32e8T + 2.47e18T^{2} \)
53 \( 1 + (1.37e9 + 1.37e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (-1.10e9 - 1.10e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (-3.14e9 + 3.14e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (3.21e9 - 3.21e9i)T - 1.22e20iT^{2} \)
71 \( 1 + 1.72e10iT - 2.31e20T^{2} \)
73 \( 1 + 1.53e10iT - 3.13e20T^{2} \)
79 \( 1 - 4.44e10T + 7.47e20T^{2} \)
83 \( 1 + (1.61e10 - 1.61e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 1.60e10iT - 2.77e21T^{2} \)
97 \( 1 - 7.16e9T + 7.15e21T^{2} \)
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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

−16.31850496691452351538282201153, −14.45994954693646001424328319890, −13.38440429291529395300576988148, −12.43887890093278237991412860234, −10.85265750250515289107193592122, −8.055682782640770383846653859027, −7.38674837004413450008298241633, −4.86266919652525414408886891838, −3.53242023440833999753727614531, −1.01294305109356082414147350480, 2.67701747207352616616659873139, 3.68988816558517949406720314529, 5.76060532789772543570671514564, 7.67203944317481421353285994794, 9.776722065859260632553988079858, 11.50552136287595374658119953173, 12.32262286972320345003627527845, 14.49218609331961073825115118109, 15.17388281584597589981332956691, 15.77726208226583979046094568844

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