Properties

Label 2-2e4-16.11-c6-0-8
Degree $2$
Conductor $16$
Sign $-0.555 + 0.831i$
Analytic cond. $3.68086$
Root an. cond. $1.91855$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.392 + 7.99i)2-s + (−15.4 − 15.4i)3-s + (−63.6 − 6.27i)4-s + (−66.8 − 66.8i)5-s + (129. − 117. i)6-s − 121.·7-s + (75.2 − 506. i)8-s − 251. i·9-s + (560. − 508. i)10-s + (−1.65e3 + 1.65e3i)11-s + (887. + 1.08e3i)12-s + (−836. + 836. i)13-s + (47.8 − 972. i)14-s + 2.06e3i·15-s + (4.01e3 + 799. i)16-s + 3.61e3·17-s + ⋯
L(s)  = 1  + (−0.0491 + 0.998i)2-s + (−0.572 − 0.572i)3-s + (−0.995 − 0.0981i)4-s + (−0.535 − 0.535i)5-s + (0.599 − 0.543i)6-s − 0.354·7-s + (0.146 − 0.989i)8-s − 0.345i·9-s + (0.560 − 0.508i)10-s + (−1.24 + 1.24i)11-s + (0.513 + 0.625i)12-s + (−0.380 + 0.380i)13-s + (0.0174 − 0.354i)14-s + 0.612i·15-s + (0.980 + 0.195i)16-s + 0.736·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.555 + 0.831i$
Analytic conductor: \(3.68086\)
Root analytic conductor: \(1.91855\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :3),\ -0.555 + 0.831i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.100437 - 0.187949i\)
\(L(\frac12)\) \(\approx\) \(0.100437 - 0.187949i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.392 - 7.99i)T \)
good3 \( 1 + (15.4 + 15.4i)T + 729iT^{2} \)
5 \( 1 + (66.8 + 66.8i)T + 1.56e4iT^{2} \)
7 \( 1 + 121.T + 1.17e5T^{2} \)
11 \( 1 + (1.65e3 - 1.65e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (836. - 836. i)T - 4.82e6iT^{2} \)
17 \( 1 - 3.61e3T + 2.41e7T^{2} \)
19 \( 1 + (5.24e3 + 5.24e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 77.1T + 1.48e8T^{2} \)
29 \( 1 + (-3.17e4 + 3.17e4i)T - 5.94e8iT^{2} \)
31 \( 1 + 3.58e3iT - 8.87e8T^{2} \)
37 \( 1 + (4.44e4 + 4.44e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 1.21e5iT - 4.75e9T^{2} \)
43 \( 1 + (6.21e4 - 6.21e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 9.96e4iT - 1.07e10T^{2} \)
53 \( 1 + (1.23e5 + 1.23e5i)T + 2.21e10iT^{2} \)
59 \( 1 + (1.06e3 - 1.06e3i)T - 4.21e10iT^{2} \)
61 \( 1 + (1.03e5 - 1.03e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (1.19e4 + 1.19e4i)T + 9.04e10iT^{2} \)
71 \( 1 - 6.03e5T + 1.28e11T^{2} \)
73 \( 1 + 1.02e5iT - 1.51e11T^{2} \)
79 \( 1 + 3.87e5iT - 2.43e11T^{2} \)
83 \( 1 + (8.45e4 + 8.45e4i)T + 3.26e11iT^{2} \)
89 \( 1 - 3.45e4iT - 4.96e11T^{2} \)
97 \( 1 + 3.31e5T + 8.32e11T^{2} \)
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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

−17.32582843637433638429439141101, −16.04590305558618682409312077587, −14.97216362344828764997862126225, −13.08450152970120103943757153918, −12.19859973011130583438470300143, −9.818459786726049055575870818176, −7.979313191331276345010626771898, −6.61375926822799670799108445929, −4.77888632533705511135443723445, −0.16079900154537075964436078345, 3.23215198616290455502743570193, 5.29177965998331309453277792264, 8.162743539938543482036713868450, 10.30008768326584818925005354703, 10.92638527045824643629486887844, 12.44486373501045684079163752693, 13.96971622636684468712757954310, 15.73022632417517667249523783785, 16.98533791374931982453820491474, 18.63577090543546748948189148696

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